Category: Notes

Newton’s Third Law of Motion Physics Notes

Newton’s Third Law of Motion:
If an object ‘A exerts a force on object ‘B, then object B must exert a force of equal magnitude and opposite direction back on object A.

This law represents a certain symmetry in nature: forces always occur in pairs, and one body cannot exert a force on another without experiencing a force itself. We sometimes refer to this law loosely as action-reaction, where the force exerted is the action and the force experienced as a consequence is the reaction.

According to Newton’s third law, “To every action, there is always an equal and opposite reaction”.

It must be remembered that action and reaction always act on different objects. The third law of motion indicates that when one object exerts a force on another object, the second object instantaneously exerts a force back on the object. These two forces are always equal in magnitude, but opposite in direction.

These forces act on different objects and so they do not cancel each other. Thus Newton’s third law of motion describes the relationship between the forces of interaction between two objects.

For example, when we placed a wooden block on the ground, this block exerts a force equal to its weight, W = mg acting downwards to the ground. This is the action force. The ground exerts an equal and opposite force N = mg on the block in upward direction. This is the reaction force.

Action weight of the body acting downwards

Illustrations of Newton’s Third Law:
Some of the examples of Newton’s third law of motion are given below:
1. A gun recoils when a bullet is fired form it: When a bullet is fired from a gun, the gun exerts a force on the bullet in the forward direction. This is the action force. The bullet also exerts an equal force on the gun in the backward direction. This is the reaction force. Due to the large mass of the gun it moves only a

little backward by giving a jerk at the shoulder of the gun man. The backward movement of the gun is called the recoil of the gun.

2. Walking: In order to walk, we press the ground in backward direction with our feet (action). In turns, the ground gives an equal and opposite reaction R, (figure (a)). The reaction R can be resolved into two components, one along the horizontal and other along the vertical. The component H = R cosθ along the horizontal, help us to move forward, while the vertical component, V = R sinθ opposes our weight, [figure (b)]

3. Swimming: While swimming, the swimmer pushes the water backward (action). The water pushes the swimmer forward (reaction) with the same force.

The swimmer pushes down and backwards against the water.

4. Flight of jet planes and rockets: Rockets move forward by expelling gas backward at high velocity. This means the rocket exerts a large backward force on the gas in the rocket combustion chamber, and the gas, therefore, exerts a large reaction force forward on the rocket. This reaction force is called thrust. It is a common misconception that rockets propel themselves by pushing on the ground or on the air behind them. They actually work better in a vacuum, where they can more readily expel the exhaust gases.

5. The flying of a bird: The wings of a bird force air a downward direction. In turn, the air gives equal and opposite reactions to the wings. The resultant reaction acts on the bird in an upward direction and makes the bird to fly upward.

Similarly, the Helicopter creates lift by pushing air down, thereby experiencing an upward reaction force. Birds and airplanes also fly by exerting force on air in a direction opposite to that of whatever force they need.

6. It is difficult to walk on sand or ice: As we press the sandy ground in the backward direction, the sarid gets pushed away and as a result, we get only a very small reaction from the sandy ground, making it difficult for us to walk. In case of ice, due to little friction between our feet and ice surface, there is hardly any forward reaction and hence we cannot walk on it.

7. Rebounding of a rubber ball: When a rubber is struck against a wall or floor, it exerts a force on wall (action). The ball rebounds with an equal force (reaction) exerted by the wall or floor on the ball.

Important Facts:

1. Forces of action and reaction act always on different bodies. Hence, they never cancel each other. Each force produces its own effect.
If we consider a pair of bodies A and B. then according to third law,
\(\vec{F}_{A B}=-\vec{F}_{B A}\)
i.e force on A by B= – force on B by A

2. If action and reaction forces were to act on the same body, their resultant would be zero.

3. It is wrong impression that action comes before reaction i. e., the action is the cause and reaction is the effect. The fact is that the two act at the same instant.

4. Newton’s third law is applicable for both kind of bodies i.e.. for bodies at rest or they are in motion.

5. Forces always occur in pairs.

6. The third law applies to all types of forces, e.g, gravitational, electric or magnetic forces etc.

Physics Notes

Impulse and Impulse-Momentum Theorem Physics Notes

Impulse and Impulse-Momentum Theorem:
A force which acts on a body for short interval of time is called impulsive force or impulse.

For Example Hitting, jumping, diving, catching etc. are all examples of impulsive forces or impulse.

An impulsive force does not remain constant, but changes first from zero to maximum and then from maximum to zero. Thus it is not possible to measure easily the value of impulsive force because it changes with time. In such cases, we measure the total effect of the force, called impulse hence, impulse is defined as the product of the average force and the time interval for which the force acts.

If \(\vec{F}\) is the value of force during impact at any time and \(\vec{p}\) is the momentum of the body at that time, then according to Newton’s second law of motion,

\(\vec{F}\) = \(\frac{d \vec{p}}{d t}\)
or \(\vec{F}\) dt = d\(\vec{p}\)

Suppose that the impact lasts for a small time t and during this time, the momentum of the body changes from \(\vec{p}_{1}\) to \(\vec{p}_{2}\) then integrating the above equation between the proper limits, we have:

It may be noted that F has not been taken out of the integration sign for the reason that varies with time and does not remain constant during impact. The integral ∫t0 F dt is measure of the impulse, when the force of impact acts on the body and from equation (1) we find that it is equal to total change in momentum of the body. Since impulse is equal to a scalar (time) times a vector (force) or equal to the change in momentum (vector), it is a vector quantity and it is denoted by I.

However if \(\vec{F}_{a v}\) is the average force (constant) during the impact, then

i. e., the change in momentum of an object equals the impulse applied to it. This statement is called
impulse-momentum theorem.
I = Δp
Dimensional Formula and Unit:
I = FΔt = [M¹L¹T^-2] |T¹]= [M¹L¹T^-1]
So, the dimensional formula of impulse is the same as that of momentum.

The SI unit of impulse are (N-s) and kg m/s.
In C.G.S. system unit of impulse are dyne-sec and g cm/s.

Force Time Curve: In the real world, forces are often not constant. Forces may build up from zero over time and also may vary depending on many factors.

Finding out the overall effect of all these forces directly would be quite difficult. As we calculate impulse, we multiply force by time. This is equivalent to finding out the area under a force time curve. For variable force the shape of the force-time curve would be complicated but for a constant force we will get a simpler rectangle. In any case, the overall net impulse only matters to understand the motion of an object following an impulse.

In the figure, the graph of change in impulsive force with the time is shown:

The force-time curve and the area between the time axis can be divided in the form of many slabs. Suppose the value of force F is considered as constant along the change in time dt, then area of slab is given by F. dt.

The total effect of the force for time t₁ to t₂
= ∫^t₀ Fdt = sum of area of all slabs o
= graph of force-time and area covered between the time axis
∵ By the impulse-momentum theorem t
Impulse I = ∫^t₀ Fdt = p₂ – p₁ = change in momentum.

Thus force-time graph and the area covered with the time axis is equal to the total change in the momentum of the body.

Applications of the Concept of Impulse:
1. While catching a ball, a cricket player lowers his hands to save himself from getting hurt:

By lowering his hands, the cricket player increases the time interval in which the catch is completed. As the total change in momentum takes place in a large time interval, the time rate of change of momentum of the ball decreases. So, according to Newton’s second law of motion, lesser force acts on the hands of player and he saves himself from getting hurt.

2. Cars, buses, trucks, bogies of train etc. are provided with a spring system (shockers) to avoid severe jerks:
When they move over an uneven road, impulsive forces are exerted by the road. The function of shockers is to increase the time of impact. This would reduce the force/jerks experienced by the rider of the vehicle.

3. An athlete is advised to come to stop slowly: After finishing a fast race an athlete is advised to come to stop slowly, so that time of stop increases and hence force experienced by him decreases.

4. In a head-on collision: Between two vehicles change in linear momentum is equal to sum of the linear momentum of the two vehicles. As the time of impact is small, an extremely large force develops which causes damage to the vehicle.

Physics Notes

Momentum and Newton’s Second Law of Motion Physics Notes

Momentum and Newton’s Second Law of Motion:
Linear momentum: Momentum of the body is the physical quantity of motion possessed by the body and mathematically, It is defined as the product of mass and velocity of the body.

As the linear momentum or simply momentum is equal to a scalar time a vector (velocity), it is
therefore a vector quantity and is denoted by p. The momentum of a body of mass m moving with velocity v is given by the relation:
\(\vec{p}=m \vec{v}\)
Dimensional formula = [M¹L¹T^-1]
Unit = Kg m/s

Suppose that a ball of mass mi and a car of mass m₂ (m₂ > m₁) are moving with the same velocity v. If p₁ and p₂ are momentum of ball and car respectively then:
\(\frac{p_{1}}{p_{2}}=\frac{m_{1} v}{m_{2} v}\)
or
\(\frac{p_{1}}{p_{2}}=\frac{m_{1}}{m_{2}}\)

As m₂ > m₁: It follows that p₂ > p₁. If a ball and a car are traveling with the same velocity, the momentum of the car will be greater than that of the ball. Similarly, we can show that if two objects of same masses are thrown at different velocities, the one moving with the greater velocity possesses greater momentum. Finally, if two objects of masses m₁ and m₂ are moving with velocities v₁ and v₂ possess equal momentum.
m₁v₁ = m₂ v₂
\(\frac{v_{1}}{v_{2}}=\frac{m_{2}}{m_{1}}\)
In case m₂ > m₁ then v₂ < v₁ i. e, two bodies of different masses possess same momentum, the lighter body possesses greater velocity.

The concept of momentum was introduced by Newton in order to measure the quantitative effect of force.
The momentum of body in terms of kinetic energy

Explanation of Newton’s Second Law:
According to Newton’s second law of motion, the rate of change of linear momentum of a body is directly proportional to the applied external force on the body, and this change takes place always in the direction of the applied force.
Let, m = mass of a body
v = velocity of the body
The linear momentum of the body
\(\vec{p}=m \vec{v}\) …(1)
let \(\vec{F}\) = External force applied on the body in the direction of motion of the body. .
Δ \(\vec{p}\) = a small change in linear momentum of the body in a small time Δt.
Rate of change of linear momentum of the body = \(\frac{\Delta \vec{p}}{\Delta t}\)

According to Newton’s second law

Where k is proportionality constant.
Taking the limit Δt → 0, the term \(\frac{\Delta \vec{p}}{\Delta t}\) becomes the derivative or differential coefficient of \(\vec{p}\) w.r.t. time t.
It is denoted by \(\frac{d \vec{p}}{d t}\)
\(\vec{F}=k \frac{d \vec{p}}{d t}\)
Where k = 1 in all the system

As acceleration is a vector quantity and mass is scalar, therefore force \(\vec{F}\) being the product of m and \(\vec{a}\) is a vector. The direction of is the same as the direction of \(\vec{a}\). Equation (4) represents the equation of motion of the body. We can rewrite equation (6) in scalar form as:
F = ma …(7)
Thus, magnitude of the force can be calculated by multiplying mass of the body and the acceleration produced in it. Hence, second law of motion gives a measure of force.

Important Facts:
1. If the applied force produces acceleration a, such that ax,ay and a2 are the magnitudes of the component of acceleration along the X-axis, Y-axis, X-axis respectively.
Then
\(\vec{F}\) = m(a_x î + a_y ĵ + a_z k̂) …(iv)
If F_x, F_y ,F_z are components of force along X-axis, Y-axis and X-axis respectively, then
\(\vec{F}\) = F_xî + F_y ĵ + F_z k̂ …….(v)

From the equation (4), (5) we have

The set of equation (8) expresses Newton’s second law of motion in component form. Three mutually perpendicular components of the force and the acceleration have to obey the set of equation (8).

As the force is equal to a scalar (mass) times a vector (acceleration), it is a vector quantity. The equation (8) is called the equation of motion of body.

In scalar form F = ma
If we know the values of m and a, the force F acting on the body can be calculated and hence second law gives a measure of the force.

2. Equal forces applied for equal time on different bodies change equal momentum.

3. Newton’s second law shows the relation between net external force and acceleration of the body.

4. Initially two bodies are in rest position, a constant force is applied for a definite time interval then the lighter body receives more speed than the heavy body because of change in momentum In both bodies are equal.
Thus, m₁v₁ = m₂v₂
If m₁ < m₂
then v₁ > v₂

Dimension and Unit of Force:
As F = ma
.-. F= [M¹] [L¹T^-2 ] = [M¹L¹T^-2]
This is the dimensional formula of force.

Unit of force:
The unit of force is Newton in the (M.K.S.) system and dynein the C.G.S. system and Poundal in (F.P.S.) system.
Definition of 1 Newton
F = ma
When m = 1kg and a = 1 m/s
then F = 1 Newton
The force which produces the acceleration of 1m/s^-2 in the 1 kg body, is equal to the 1 Newton.
1 Newton = 1 kg m/ s²

In C.G.S. System
F = ma
If m = 1 g and a = 1 cm/s²
then F = 1 dyne
Thus 1 dyne is that force that produces 1 cm/s acceleration in the mass of 1g body.

Physics Notes

Newton’s First Law of Motion or Law of Inertia Physics Notes

Newton’s First Law of Motion or Law of Inertia:
Sir Isaac Newton’s first law of motion describes the behaviour of a massive body at rest or in uniform linear motion i. e., not accelerating or rotating.

The first law states, “A body at rest will remain at rest and a body in motion will remain in motion unless in the abesence of external force.” It is also called the law of inertia.

This simply means that things can not start, stop or change direction all by themselves. It requires some force acting on them from the outside to cause such a change. While this concept seems simple and obvious to us today, in Newton’s time it was truely revolutionary.

Following inferences can be easily drawn from Newton’s first law of motion:

• If a body is at rest, then so as to set it in motion an external force has to be applied on it.
• If a body is moving with a constant speed along a straight line. Then in order to increase or decrease its speed, an external force has to be applied in the direction of motion or opposite to the direction of motion.
• If a body is moving with a constant speed along a straight line, then in order to change its direction of motion an external force has to be applied in a direction normal to the direction of motion.
  From the above discussion, it follows that the first law of motion is simply the law of inertia.

Application of Law of Inertia:
1. A ball thrown upward in a train moving with a uniform velocity, returns to the thrower:
Because during the upward and the downward journey, due to inertia, the ball also moves along the horizontal with the velocity of the train. Hence, it covers the same horizontal distances as the train does and the ball returns to the thrower.

2. When a running car suddenly stops, the rider falls forwards: Because the lower part of the body of the rider (which is in contact with the car) comes to rest. But due to of inertia, the upper part of the body tends to keep moving. As a result of it the rider falls in the forward direction.

3. When a car suddenly starts moving, the rider falls in backward: Because the lower part of the body of the rider (which is constant with the car) comes in motion, but due to the inertia the upper part tends to be at rest. Hence, the rider falls in backward.

4. Dust is removed from a having carpet by beating it with a stick: The carpet comes into motion, but because of inertia the dust particles remain at rest, due to this, the dust particles get removed from the carpet.

5. An athlete runs for a certain distance before taking a long jump: When an athlete runs some distance the velocity acquired due to inertia is added to the velocity of the athlete at the time to jump. Since length of the jump depends upon the initial velocity, athlete is likely to jump a longer distance by doing so.

Physics Notes

Concept of Force and Inertia Physics Notes

Concept of Force and Inertia:
Force: It is a push or pull that either changes or tends to change the state of rest or of uniform motion of a body or the direction of motion of a body.

It is our common observation that an object such as a chair lying in a room or a vehicle parked outside the house remains at rest unless a push or a pull is given to it. Such an object cannot move at its own. In other words force has to be applied in order to move an object at rest.

Also if an object is moving along a straight line with some velocity, it is found that force is required to change its direction of motion or the magnitude of its velocity. In other words force is an agent which causes acceleration. However, in certain cases the acceleration caused by the force may be practically zero. For example, if we push a stationary truck, it may not move. In such cases we say that the force merely tends to cause acceleration zero.

Hence, force is a push or pull which produces or tends to produce a motion in a body at rest, stops or tends to stop a body which is in motion, increase or decreases the magnitude of the velocity of the moving body or changes the direction of motion of the moving body.

To describe the applied force on a body, the following facts are need to be determined.

1. The magnitude of applied force.
2. Direction of applied force.
3. Point of position of applied force.

Inertia: In the absence of external force the inability of a body to change its state by itself is called Inertia.
It is the property of an object which resists the change in the state of linear motion. It is equal to the mass of the object.

As said above an object such as a chair lying in the room remains at rest unless an external force is applied on it. It can be set into motion only by applying force on it. In other words, if we do not apply a force on the stationary chairs, it will not move at all. This inherent property of the objects that they do not change their state, unless acted upon by an external force is called Inertia.

It was the general belief centuries before; that force is necessary even to keep a body in motion with uniform velocity. With his simple experiments with inclined planes. Galileo proved that no force was required to keep a body moving with uniform velocity provided friction is not present. Therefore, it is the inherent property of the object that they remain moving with a constant speed along a straight line unless acted upon by an external force. Thus, it is due to inertia that a body cannot change its state of rest or a uniform motion,by itself. It is sometimes also called the Galileo Law of Inertia.

Kind bf Inertia:

• The inertia of rest: It is defined as the inability of a body to change by itself its state of rest. This means a body at rest remains at rest and can not start moving by its own. This is called inertia of rest.
• The inertia of motion: It is defined as the inability of a body to change itself its state of uniform motion. This means that body in uniform motion can neither accelerate nor retard by its own. This is called inertia of motion.
• The inertia of direction: It is defined as the inability of a body to change by itself the direction of motion.
• Force produces acceleration and retardation in moving body.
• Inertia is not a physical quantity, it is only the inherent property of a body which depends upon the mass of the body.
• Inertia has no unit and dimension.
• Two bodies of same mass have equal inertia because the inertia only depends on the mass. It does not depend upon the velocity and size of the body.
• Inertia means resistance to the change in its state.

Physics Notes

Laws of Motion Physics Notes

we have learnt to describe the motion of an object in terms of its displacement, velocity and acceleration. Now, the important question arises: what makes an object move? Or what makes a ball roll along the ground to come to halt? We know from our everyday experience that we need to push or pull an object if we want to change its position.

The Greek thinker Aristotle (384 B.C. – 322 B.C.) held the view that if a body is moving, something external is needed to keep it moving.

However, there are some situations where the cause behind an action is not visible. For example: Why raindrops fall to the ground? Why does the Earth move around the Sun?

In this chapter, we will learn the basic laws of motion and discover the forces that cause motion. The concept of force discussed in this chapter will be useful in different branches of Physics.

Newton demonstrated that force and motion are closely connected. The laws of motion are fundamental and make us understand everyday phenomenon.

→ Force: It is a push or a pull, which either changes or tends to change the state of rest or of uniform motion of a body. It is a vector quantity and is devoted by \(\vec{F}\).

→ Inertia: It is the property of matter in which an object that is at rest wants to remain at rest, and object that is moving wants to remain moving in a straight line unless another force acts upon it.
In other words inertia is the resistance of any physical object to any change in its state of motion. This includes changes to the object’s speed, direction or state of rest.

→ Momentum: It is the quantity of motion contained in a body and is measured as the product of the mass of the body and its velocity. It is a vector quantity and is denoted by \(\vec{P}\) mathematically, \(\vec{p}\) = m \(\vec{v}\)

→ Newton’s first law of motion: It states that an object at rest, stays at rest and an object in motion stays in motion with the same speed in the same direction unless acted upon by an unbalanced force. It is sometimes referred to as the law of inertia.

→ Newton’s second law of motion: It states that the time rate of change of momentum of a body is directly proportional to the external force applied on it and the change in momentum takes place in the direction of force.
Mathematically,

Absolute unit of force: In SI, the absolute unit of force is Newton (N):
1 Newton (1N) = 1 kg 1m/ s² =1 kgm/s²

→ Newton’s third law of motion: It states that to every action, there is an equal and opposite
reaction. Mathematically, \(\overrightarrow{F_{A B}}=-\vec{F}_{B A}\)

→ Impulse: It is defined as the product of the average force acting during the impact and the time for which the force lasts. In Classical Mechanics, the impulse is the integral of a force F, over the time interval, t, for which it acts. Since force is a vector quantity, the impulse is also a vector in the same direction.

It is also equal to the total change in momentum produced during the impact.

If \(\vec{F}_{a v}\) is the average force acting during the impact then \(\vec{I}=\vec{F}_{a v} t=\overrightarrow{p_{2}}-\overrightarrow{p_{1}}\)
Unit: SI, unit of impulse is Ns or kg m/s.

→ Law of conservation of linear momentum: It no external force acts on a system, then its total linear momentum remains conserved.
Mathematically, \(m_{1} \overrightarrow{u_{1}}+m_{2} \overrightarrow{u_{2}}=m_{1} \overrightarrow{v_{1}}+m_{2} \overrightarrow{v_{2}}\) provided Fext =0.

Linear momentum depends on frame of reference but law of conservation of linear momentum is independent of frame of reference.
Newton’s law of motion are valid only in inertial frame of reference.

→ Rocket Propulsion: Rocket is an example of variable mass, following law of conservation of momentum.
(a) Thrust on rocket at any instant: F = -v\(\frac{d m}{d t}\)
Where v = exhaust speed of the burnt gases and \(\frac{d m}{d t}\) = rate of gases combustion of fuel
(b) Velocity of rocket at any instant is given by
v = v₀ + vg loge\(\left(\frac{m_{0}}{m}\right)\)

where v₀ = initial velocity of rocket
m₀ = initial mass of the rocket
m = present mass of the rocket

If the effect of gravity is taken in to account, then speed of rocket ,
v = v₀ + vg loge\(\left(\frac{m_{0}}{m}\right)\) – gt

→ Friction: A force acting on the point of contact of the objects, which opposes the relative motion is called friction. It acts parallel to the contact surfaces.

Frictional forces are produced due to inter-molecular interaction acting between the molecules of the bodies in contact. Friction is of three types:
(a) Static friction: It is an opposing force that comes into play when one body tends to move over the surface of the other body but the actual motion is not taking place.
Static friction is a self-adjusting force that increases as the applied force is increased.

(b) Limiting friction: It is the maximum value of friction when the body is at the verge of starting motion.
Limiting friction (f_s)max = μ_sN = μsmg
Where, μ_s = coefficient of limiting friction and N = normal reaction
Limiting friction does not depend on the area of contact surfaces but depends on their nature i. e., smoothness or roughness.

(c) Kinetic friction: If the body begins to slide on the surface, the magnitude of the frictional force rapidly decreases to a constant value f which is called kinetic friction.
Kinetic friction f_k = μ_kV
= μ_kmg
where μk = coefficient of kinetic friction and
N = normal force

Kinetic friction is of two types:

1. Sliding friction
2. Rolling friction

As rolling friction < sliding friction,
Therefore it is easier to roll a body than to slide.

→ Centripetal force: An external force required to make a body move along the circular path with uniform speed is called centripetal force.
It acts along a radius and towards the centre of the circular path.
Mathematically: Centripetal force
= \(\frac{m v^{2}}{r}\) = mω²r
= 4mπv²r

→ A vehicle taking circular turns on a level road: If the coefficient of friction between the tyre and the road is p, the maximum velocity which vehicle can take a circular turn of radius r is given by
vmax = \(\sqrt{\mu \mathrm{rg}}\)

→ Banking of tracks (roads): So that vehicles can move on a curved track of radius r with a maximum speed v, the track is banked through an angle θ given by
v² = rgtanθ
or tan θ = \(\frac{v^{2}}{r g}\)
The maximum permissible speed of a vehicle on a banked road at angle θ is given by
v_max = \(\left[\frac{r g(\mu+\tan \theta)}{1-\mu \tan \theta}\right]^{1 / 2}\)
where µ = coefficient of the friction between the road and tyres of the vehicle.

→ Motion in a vertical circle: For a body of mass ‘m’ to just loop a vertical circle of radius r,
(a) Minimum velocity of the body at the lowest point;
v₁ = \(\sqrt{5 g r}\)
(b) Velocity at the highest point, u² = \(\sqrt{g r}\)
(c) Tension in the string at the lowest point, T = 6 mg

→ Concurrent Forces: The forces acting at the same point are called concurrent forces.

→ Equilibrium of concurrent forces: A number of concurrent forces acting on a body are said to be in equilibrium, if the resultant of these forces in zero or if the concurrent forces can be represented completely by the sides of a closed polygon taken in the same order
Mathematically: \(\vec{F}+\vec{F}_{2}+\vec{F}_{3}+\ldots+\vec{F}_{n}\) = 0

→ Free body diagram: A diagram for each body’ of the system depicting all the forces on the body by the remaining part of the system is called the free body diagram.

→ Force: Is an external agency which changes or tends to change the state of a body.

→ Inertia: Is the property of an object which resist to the change.

→ Momentum: The product of mass and velocity of a body which is moving.

→ Newton: The force which produces the acceleration of 1m/s 2in a mass of 1 kg, is equal to one Newton force.

→ Impulse: The total effect of force on the motion of a body is called impulse.

→ Friction: The force that resists the relative motion of solid surfaces, fluid layers and material elements that slide against each other is known as friction.

→ Concurrent forces: Are two or more forces whose lines of action intersect at a common point.

Physics Notes

Time of Flight and Horizontal-range Physics Notes

Time of Flight:
It is the total time taken by the projectile when it is projected from a point and reaches the same horizontal plane or the time for which the projectile remains in the air above the horizontal plane.

It is denoted by T.
As the motion from the point O to A and then from point A to B are symmetrical, the time of ascent (for the journey from point O to A) and the time of descent (for the return journey from A to B) will be each equal to T/2.

At the highest point A, the vertical component of velocity of the object becomes zero. Taking vertically upward motion of the object from O to A, we have

The time of flight is independent of the horizontal component of velocity. The faster a projectile is thrown up, the longer it will stay is the air.

Maximum height of a projectile:
It is the maximum vertical height attained by the object above the point of projection during its flight. It is denoted by H.

Taking the vertical upward movement of the object 122 (B)from O to A, we have:

1. The maximum height is independent of the horizontal component of velocity. The faster a projectile is thrown upwards, the higher it will go in an upward direction, i.e., the longer it will resist the downward pull of gravity.

2. Both time of flight and maximum height depends upon the vertical component of velocity, thus the relation between them can be expressed as
\(\frac{H}{T^{2}}=\frac{g}{8}\)

Horizontal-range:
It is the horizontal distance covered by the object between its point of projection and the point of hitting the ground. It is denoted by R.

Obviously, the horizontal range R is the horizontal distance covered by the projectile with the’ uniform velocity u cos θ in a time equal to the time of flight.

The angle of projection for maximum range:
The value of the horizontal range depends upon the angle of projection O. Therefore, horizontal range R will be maximum if
sin 2θ = maximum = 1 = sin90°
or 2θ = 900
or θ = 45°
∴ Maximum horizontal range, Rmax = \(\frac{u^{2}}{g}\) …..(7)

1. The horizontal range depends upon both the horizontal and vertical components of velocity.
2. For a specified speed of projection, the range will maximum at an angle of projection equal to 45°
3. Projectile moving at equal speed, the range will be equal when both projectiles have a complementary angle of projection. it means θ or 90-θ

Physics Notes

Projectile Motion Physics Notes

Projectile Motion:
Projectile motion is a two-dimensional motion in which an object or particle is thrown upward at an angle to the horizontal and it moves along a curved path due to the action of gravity. The only force of significance that acts on the object is gravity, which acts downward.

The object projected into space or air is called a projectile and the path followed by the projectile is called a trajectory.

Following are a few examples of the projectiles:

• A bullet fired from a gun.
• A javelin or hammer was thrown by an athlete.
• A football kicked in air
• A piece of stone is thrown in any direction.
• A jet of water ejecting from a hole near the bottom of the water tank.
• An arrow was released from the bow.
• A missile deployed from a military aircraft from level flight.

Velocity and Acceleration in Projectile Motion
Projectile motion is a planar motion is which at least two position coordinates change simultaneously.

1. The motion of a projectile is a two-dimensional motion so, it can be discussed in two parts. Horizontal motion and vertical motion. These two motions take place independently of each other.
2. The velocity of the projectile can be resolved into two mutually perpendicular components; the horizontal component and the vertical component.
3. Acceleration changes velocity. If acceleration in a particular direction is zero then velocity in that direction remains the same. Thus, in projectile motion the horizontal component of velocity remains unchanged throughout the motion. The horizontal motion is a uniform motion.
4. The force of gravity continuously affects the vertical components so the vertical motion is a uniformly accelerated motion.

Projectiles can be thrown in various ways; on level ground, from a high tower to ground, from an airplane etc.

To study the motion of a projectile we assume that:

• There is no friction due to air
• The effect due to the curvature of the earth is negligible.
• The entire trajectory is near the surface of the earth.

Path of a projectile:
Let OX be a horizontal line on the ground and OY be a vertical line: O is the origin for X and Y-axis.

Consider that a projectile is fired with velocity u and making an angle θ with the horizontal from the point ‘O’ on the ground [figure]

The velocity of projection of the projectile can be resolved into the following two components

• u_x = u cos θ, along OX
• u_y = u sin θ, along OY

As the projectile moves, it covers distance along the horizontal due to the horizontal component u cos0 of the velocity of projection and along vertical due to the vertical component u sin0. Let that any time t, the projectile reaches the point P, so that its distances along the X and Y-axis are given by x and y respectively.

Motion along horizontal direction: If we neglect the friction due to air, then the horizontal component of the velocity i. e., u cos0 will remain constant. Thus

Initial velocity along the horizontal, u_x = u cos θ
Acceleration along the horizontal, a_x = 0
The position of the projectile along the X-axis at any time t is given by

Motion along vertical direction:
The velocity of the projectile along the vertical goes on decreasing due to the effect of gravity

Initial velocity along vertical, u_x = u sin θ
Acceleration along vertical, a_y = -g

The position of the projectile along the Y-axis at any time t is given by

This is an equation of a parabola. Hence the path of a projectile projected at some angle with the horizontal direction is a parabola.

Physics Notes

Two Dimensional and Three Dimensional Motion Physics Notes

Two Dimensional and Three Dimensional Motion:
Earlier we have studied that on the basis of frame of reference, motion is divided into three types of motion. According to which if there is a change in two coordinates of the frame of reference of a moving object ‘ as time passes then it is called two-dimensional motion, and if all the three coordinates change with time it is called three-dimensional motion.

Displacement, Velocity, and Acceleration of a Particle in Two Dimensional Motion and Their Vector Representation

To study about displacement, velocity, and acceleration of a particle in two-dimensional motion we study the the motion of the particle in the XY axis of the reference frame. Assume that at any time interval t₁ the position of the particle is A₁ whose position vector is \(\vec{r}_{1}\) and at any time t₂ the position of the particle is A₂ whose position vector is \(\vec{r}_{2}\). Therefore, the vector representation of points A₁ and A₂ is following;

The motion of a particle in a plane

\(\vec{r}_{1}\) = x₁î + y₁ĵ …….(i)
Where î and ĵ are unit vectors in the direction of x and y-axis representing directions respectively.
and \(\vec{r}_{2}\) = x₂î + y₂ĵ …(2)

Again, because the particle is displaced from A₁ to A₂. Hence, according to the diagram if displacement is A₁ then according to the triangle rule in the triangle OA₁A₂.

Therefore, the displacement equation would be given as equation (3) in two-dimensional motion.

According to Vector Algebra, the resultant is;
Δr = \(\sqrt{(\Delta x)^{2}+(\Delta y)^{2}}\)
This equation (3) is the result of displacement and the direction of displacement vector Δr would be according to the figure.

To calculate the average velocity of the particle according to the definition of average velocity;

Hence, average velocity would be given as equation (5) and the resultant of average velocity by
Vector Algebra would be;
v = \(\sqrt{\left(v_{x}\right)^{2}+\left(v_{y}\right)^{2}}\) ………(6)
given by equation (6) and the direction of it would be in the direction of \(\overrightarrow{\Delta r}\). The instantaneous velocity of this particle at any instant t by the definition of instantaneous velocity would be;

Hence, the instantaneous velocity is given by the Eqn. (7). Resultant of instantaneous velocity by Vector Algebra.
v = \(\sqrt{v_{x}^{2}+v_{y}^{2}}\) ……….(8)

will be given by Eqn. (3) and direction would be in the direction of the tangent at the point; which is the position of the particle at time t.

Distribution of velocity in its components

Suppose it makes an angle θ with the x-axis; then the direction would be as shown in the figure and by the equation;

If the object is executing accelerated motion and in the time interval Δt the change in velocity is Δu; then by the definition of average acceleration, the acceleration of the particle will be;

This is the equation of average acceleration. The result of this by Vector Algebra is given by
a = \(\sqrt{\left(a_{x}\right)^{2}+\left(a_{y}\right)^{2}}\) ……..(11)
will be given by equation (11) and the direction of it would be in the direction of Δv.

To calculate instantaneous acceleration by definition;

Equation (12) is the vector equation of instantaneous acceleration; and its result;
a = \(\sqrt{\left(a_{x}\right)^{2}+\left(a_{y}\right)^{2}}\) ……(13)
will be given by (13) and its direction would be in the direction of v. All the various equations above explain displacement, velocity and acceleration in two-dimensional motion.

Displacement, Acceleration, Velocity of a Particle in Three Dimensional Motion and Their Vector Representation

If the object is moving in space then the object is in three-dimensional motion; and its position, velocity, acceleration, and other components along with x and y-axis would also change in the direction of the z-axis.
The position vector, velocity vector and acceleration vector of three dimensional motion are given below;
r = xî + yĵ + zk̂ = \(\vec{x}+\vec{y}+\vec{z}\) ……(14)
v = vxî + vyĵ + vzk̂ = \(\overrightarrow{v_{x}}+\overrightarrow{v_{y}}+\overrightarrow{v_{z}}\) …(15)

Physics Notes

Relative Motion Physics Notes

Relative Motion:
The relative velocity of the first body with respect to another body, when both are in motion, is the rate at which the first body changes its position with respect to another body.

The concept of relative motion or relative velocity is all about understanding the frame of reference. A frame of reference can be thought of as the state of motion of the observer of some event. For example, if you are sitting on a lawn chair watching train travel past you from left to right at 50 m/s. You would consider yourself in a stationary frame of reference from your perspective, you are at rest, and the train is moving. Further, assuming you have tremendous eyesight, you could even watch a glass of water on a table inside the train move from left to right at 50 m/s.

An observer on the train itself, however, sitting beside the table with a glass of water, would view the glass of water as remaining stationary from their frame of reference, because that the observer is moving at 50 m/s, and the glass of water is moving at 50 m/s, the observer on the train sees no motion for the glass of water.
Due to different frame of reference and their positions, the motion of an object relative to different reference frames can be similar or different.

Position of a point in a different frame of references

According to the figure, let us discuss the two reference frames S and S’ whose origins O and O’ are relative and parallel. In the reference frame S,r is the vector of point P relative to O and in the reference frame S’, the vector is r0 with relative to O’. Then, in the reference frame S’, the vector r’ of point P relative to O’ will be given as;
r’ = r – r₀ …(1)
Differentiating the above equation with respect to t,
\(\frac{d r^{\prime}}{d t}=\frac{d r}{d t}-\frac{d r_{0}}{d t}\)
or v’= v – v₀ …(2)
Here, v0 is the velocity in the reference frame S’ relative to S.

Equation (2) depicts the relationship between the velocity vectors of the references S and S’.
Again differentiating equation (2) with respect to t.
\(\frac{d v^{\prime}}{d t}=\frac{d v}{d t}-\frac{d v_{0}}{d t}\)
or a’ = a – a₀ …(3)
Here, a0 is the acceleration of reference frame S’ relative to S.

If a₀ = 0 i. e., if frame S and S’ are moving relative to one another with definite velocity then; a’= a

Hence, in this state, the acceleration of any particle would be the same in both the frame of reference.

Considering in one-dimensional motion; if two objects A and B are moving in the direction of the x-axis with definite velocity v_A and v_B. At any instant the position of the objects would be x_A and x_B respectively, then the position of B relative to A would be;
x_BA = x_B – x_A ……..(4)
and B’s velocity relative to A would be;
v_BA = v_B – v_A ……….(5)

Note: Relative velocity is used to describe the motion of airplanes in the wind or moving boats through water etc. This velocity is computed according to the observer inside the object. This can be computed by introducing an intermediate frame of reference, in simpler words, this can be the vector sum of the velocities.

For example, the velocity of object A with respect to reference frame C would be written as V_AC. Even if you don’t know the velocity of object A with respect to C directly, by finding out the velocity of object A with respect to some intermediate object B, and the velocity of object B with respect to C, you can combine your velocities using vector addition to obtain:

\(\overrightarrow{v_{A C}}=\overrightarrow{v_{A B}}+\overrightarrow{v_{B C}}\)
This sounds more complicated than it actually is. Let’s look at how this is applied in few examples:

Physics Notes

Equations of Motion for Uniform Accelerated Motion Physics Notes

Equations of Motion for Uniform Accelerated Motion:
In uniform acceleration, magnitude and direction of an object always remains constant. This type of motion is called uniform accelerated motion.

For uniform accelerated motion, the graph between velocity and time is a straight line and its slope is equal to the acceleration.

The area under the velocity-time graph expresses the distance or displacement.

For the study of uniform accelerated motion, the equation which shows the relation among velocity, time, distance and acceleration is called the equation of motion.

There are three equations of motion,
1. Graphical method: Consider an object moving along a straight line with uniform acceleration a. Let u be the initial velocity of the object at time t = 0 and u be the final velocity of the object at time t. Let s be the distance travelled by the object in time t.

Equation of Motion by the Graphical Method

The Velocity-time graph of this motion is a straight line PQ, as shown in the figure.
where, OP = u = RS
OW = SQ = v
and OS = PR = t
1. For the first equation of motion: We know that the slope of velocity-time graph of uniformly accelerated motion represents the acceleration of the object.
i.e., Acceleration = slope of the velocity-time graph PQ
or a = \(\frac{Q R}{P R}=\frac{Q R}{O S}\)
a = \(\frac{S Q-S R}{O S}=\frac{v-u}{t}\)
or v – u = at
or v = u + at …….(i)
This is the first equation of uniform accelerated motion.

2. Second equation of motion: We know that the area under the velocity-time graph for a given time interval represents the distance covered by the uniformly accelerated object in that interval of time.

Distance (displacement) travelled by the object in time t is :
S = area of trapezium OSQP
= area of rectangle OSRP + Area of triangle PRQ
or S = OS × OP+ \(\frac{1}{2}\) × PR × RQ
(Area of rectangle = Length × Breadth)
(Area of triangle = \(\frac{1}{2}\) × Base × Height)
= t × u + \(\frac{1}{2}\) × t × (v – u)

(From the first equation of motion v – u = at)
= ut + \(\frac{1}{2}\) × t × at
Thus, S = ut + \(\frac{1}{2}\) at² ……..(iii)
This is the second equation of uniform accelerated motion.

3. Third equation of motion : Distance travelled by the object in time interval t is
s = area of trapezium OSQP
= \(\frac{1}{2}\) (OP + SQ) × OS
OP = SR
= \(\frac{1}{2}\)(SR + SQ) × OS …(iii)

Acceleration, a = slope of the velocity-time graph PQ

This is the third equation of uniform accelerated motion.

2. Calculus method:
1. Velocity-time relation: These equations can also be derived from the calculus method. From the definition of acceleration;
a = \(\) or dv=adt

Integrating it within the condition of motion (i.e.) when the time changes from 0 to t, velocity changes from u to u, we get

This is the first equation of motion.

2. Distance time relation : The instantaneous velocity of an object in uniformly accelerated motion is given by
v = \(\frac{d x}{d t}\) or dx = vdt
dt
Now v = u + at
∴ dx =(u + at)dt …(vii)
Let the displacement of the object from the origin of position-axis is x0 at t = 0 and x at t = t, integrating both the sides of the equation (vii) within proper limits, we have,

or x – xo = u(t – 0) + a(\(\frac{t^{2}}{2}\) – 0]
or x—x0 = ut + \(\frac{1}{2}\)at …(viii)
If x – x0 = S = distance covered by the object in time t, then from eq. (viii)
s = ut + \(\frac{1}{2}\)at2 …(ix)

This is the second equation of uniform accelerated motion.

3. Velocity-displacement relation
The instantaneous acceleration is given by

Let u and v be the velocity of the object at positions given by displacements x₀ and x.

Integrating the above equation (x) with the condition of motion, we get

If x – x₀ = s = the distance covered by the object in time t, then from eq. (xi),
\(\frac{1}{2}\)(v² – u²) = as
or v² – u² = 2as
or v² = u² + 2as …..(xii)
This is the third equation of uniform acceleration motion

Distance travelled by an object in nth second: We know that the distance travelled by an object iii time t is
s = ut + \(\frac{1}{2}\)at² ……(i)
in n sec; s = un + \(\frac{1}{2}\)an² …….(ii)

Similarly distance travelled in (n – 1) sec

Kinetic Equation of Motion under Gravity:
When an object falls freely under the effect of gravity, it is accelerated towards the centre of the earth with an acceleration of 9.8 m/sec² (or 980 cm s^-2) called the acceleration due to gravity (g). The motion of an object falling freely under gravity is, thus, a case of motion with uniform acceleration.

The kinetic equation of motion under gravity can be obtained by replacing ‘a’ by ‘g in the of motion [obtained earlier. Accordingly, the kinematic: equations of motion under gravity are as below:

• v = u ± gt
• S = ut ± \(\frac{1}{2}\)gt²
• v² = u² ± 2gs
• S_n = u ± \(\frac{1}{2}\) g(2n – 1)

For upward motion: a = g take
For downward motions: a = +g

In the above expression, resistance due to air has been neglected. Further, when an object, falls freely, its initial velocity u is zero and the value of ‘g is positive. On the other hand, when an object is thrown up against gravity, it will rise till its final velocity u becomes zero. In this case, the value of ‘g is negative.

Physics Notes