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

Graphical Representation of Motion Physics Notes

Graphical Representation of Motion:
1. A graphical representation is a pictorial representation of the relation between two sets of data of which one set is of a dependent variable and other set is of independent variables. Now here dependent variable is shown on Y-axis and independent variable is shown on X-axis.

2. To describe the motion of an object we can use line graphs. In this case line graph shows dependence of one physical quantity, such as distance (position), velocity, acceleration, on the another quantity such as time.

3. Now the slope of graph and area from figure provides other physical quantities. The slope of graph is calculated by tanθ which is the ratio of physical quantities taken on Y-axis to the X-axis. While by the area of figure, the product of quantities taken from the X axis and Y-axis which are dependent upon the figure and its shape.

Now here we showed the three types of graphical representation of motion.
Dependent Quantities

Independent Quantity

1. Position-time graph
2. Velocity-time graph
3. Acceleration-time.

Position-Time Graph:

1. The change in the position of an object with time can be represented on the position-time graph.
2. In this graph, time is taken on the X-axis and position is taken along the Y-axis.
3. The Position-time (P – T) graph of a moving body can be used to calculate the speed of the body as they specifically represent velocity.
4. The slope of the tangent at any point of position¬time graph denotes the instantaneous velocity of the object at that point or tan θ = v

(a) When the object is at rest
Its position will not change with time. Let the object be stationary at position x(t) = x0 from the origin. Then the position (x) time (t) graph for the stationary object is a straight line AB parallel to the time-axis.

Object at rest
For example: (i) A train standing at the railway track or line.
(ii) A bus standing on the road side.
Here θ = 0°
then tan θ° = v
so v = 0 {v ∴ velocity}

(b) When the object moves with a uniform motion:
Uniform motion: Uniform motion is defined as equal displacement occurring during successive equal time periods (some times called constant velocity).

The graph of position time for uniform motion in a straight line that the slope or velocity is constant.

(a) When the object is moving with a positive velocity then the slope of the position-time graph will be tan 0 and is positive then v is constant but positive and acceleration (a) = 0.

Slope of tan θ = \(\frac{y \text { axis }}{x \text { axis }}=\frac{x}{t}\) = velocity
velocity = v = constant
but a = \(\frac{d v}{d t}=\frac{d}{d t}\) (const)
a = 0

(b) When the object is moving with a negative velocity then the slope of position time graph will be negative. Hence θ = constant but θ > 90°, negative tan θ then v is constant but acceleration (a) = 0.

(c) When the object moves with a variable velocity or Non-Uniform Motion: Figure (a) depicts the non-uniform motion in which the graph is not a straight line in the position-time graph. When we calculate the slope at two points P and Q in the graph we find out that the slope at Q is more than the slope at P [tanθ₂ > tanθ₁]. Hence, the velocity at Q is more than the velocity at P.

In this way, the velocity of the particle is increasing with time. Hence, the acceleration of the particle will be positive. Similarly, figure (b) shows the non-uniform motion in the position-time graph. But here when the slopes are calculated at points P and Q; the slope at Q is less than the slope at P; which shows the decreasing velocity and depicts negative acceleration.

Non Uniform Motion

Velocity-Time Graph:
With the help of a velocity-time graph we get the knowledge about the acceleration and displacement of the particle.
1. Uniform velocity motion: Consider that an object is moving with uniform velocity v. Since the object is in uniform motion, the magnitude of it’s velocity at t = 0, t = 1s, t = 2s …. will always be v therefore graph between time and the velocity of the object will be as shown in the figure

Displacement of the object in a given time interval
Consider the two points A and B on the v – t graph corresponding to instants t₁ and t₂ respectively.
Then
Area ABB’A’= v(t₂ – t₁) …(1)

As given by equation (1), the displacement of the object in the time interval between t₁ and t₂ is
x₂ – x₁ = v(t₂ – t₁) ….(2)

It means the displacement of an object in the time interval (t₂ – t₁) is numerically equal to the area under velocity-time graph between the instant t₁ and t₂.

Note: It may be pointed out that this geometrical method of finding the displacement of an object holds good even in the case, when the object is moving with negative velocity. In such a case, the area below the velocity-time graph is taken as negative and corresponding to this, the displacement will also be negative.

2. Constant/uniform acceleration motion:
The slope of a velocity slope of the graph – tanθ – y/x time graph represents the acceleration of the object. So, the value of the slope at a particular time represents the acceleration of the object at that instant.
The slope of a velocity-time graph will be given by the following formula.
Slope of graph = tanθ = y/x time = \(\frac{v_{2}-v_{1}}{t_{2}-t_{1}}=\frac{\Delta v}{\Delta t}\)

Since \(\frac{\Delta v}{\Delta t}\) is the definition of acceleration, the At the slope of a velocity-time graph must be equal to the acceleration of the object.
Slope = acceleration

This means that when the slope is steep, the object will be changing velocity rapidly. When the slope is shallow, the object will not be changing its velocity as rapidly. This also means that acceleration will be negative if the slope is negative. Directed downwards the acceleration will be negative and if the slope is positively directed upwards the acceleration will be positive.

Uniform Acceleration Motion

3. Non-Uniform Acceleration Motion: In the figure, the velocity-time graph shows not a straight line motion. When we calculate the slope at two points P and Q in the graph we find out that the slope at Q is more than the slope at P which means that the acceleration at Q is more than the acceleration at P. In this way this graph shows non-uniform motion whose acceleration is increasing. Similarly, figure (b) shows the non-uniform motion in the

Non Uniform Acceleration Motion

velocity-time graph. But when the slopes are calculated at points P and Q, the slope at Q is less than the slope at P; which shows that the acceleration is decreasing.

Circular Sine Velocity-Time Graph:
In the figure, the velocity-time graph is a line graph. This type of motion generally depicts simple harmonic motion. In this type of motion velocity changes with time in the form of a sine function. By this graph we can easily understand positive and negative displacement. In the given graph from O to T and 2T to 3T time interval velocity is positive. Whereas, T to 2T time interval velocity is negative and the area (S₂) in the graph for this interval is also negative. Therefore, displacement from 0 to 3T time interval is:

Circular Sine Velocity Time Graph
S = S₁ – S₂ + S₃
Whereas distance covered in the time interval in 0 to 3 T is
S = S₁ + S₂ + S₃

Acceleration-Time Graph:
Similar to position-time graph and velocity-time graph acceleration-time graph also gives various information regarding the motion of a particle. In figure (a) acceleration is constant with time which depicts uniform accelerated motion. In figure (b) acceleration is changing with time which shows non-uniform accelerated motion. The change in the acceleration is a straight line or not in straight-line motion both shows non-uniform accelerated motion.

(a) Uniform Acceleration Motion

The area between the two-time intervals in the acceleration-time graph tells us about the change in velocity. As in the figure the shaded area between the time interval t₁ and t₂ tells about the change in velocity;
Δv = Area of shaded part.

(b) Non-Uniform Acceleration Motion

Physics Notes

Acceleration Physics Notes

Acceleration:
Generally, the velocity of a moving object changes with time. Sometimes the magnitude of velocity increases and sometimes it decreases. Sometimes the magnitude remains constant but the direction changes as in circular motion. The rate of change in velocity is defined as acceleration.

Therefore, “The rate of change of velocity of an object with respect to time is known as acceleration.”

In terms of the formula:
Acceleration = \(\frac{\text { Change in Velocity }}{\text { Time Taken }}\)
\(\vec{a}=\frac{\Delta \vec{v}}{\Delta t}\)

The unit of acceleration in M.K.S. system is meter/second 2 (m/s²) Its dimensional formula is [M⁰L¹F²]. Acceleration is a vector quantity. Similar to velocity it is also divided as follows:

Types of acceleration:
1. Average Acceleration: “The ratio of the total change in velocity to the total time taken is called average acceleration”.

If Δv is the change in velocity in it time interval, then;

Average acceleration (a) = \(\frac{\text { Total Change in Velocity }}{\text { Total Time Taken }}\)
= \(\left[\frac{\Delta v}{\Delta t}\right]\)

2. Instantaneous Acceleration:
Instantaneous acceleration is defined as “acceleration at any given point or at any instant of time.” If at Δt time interval velocity is Δv then according to the above definition; to calculate instantaneous acceleration
Δt → 0. Hence,

Here, \(\frac{d v}{d t}\), differentiation of v with respect to time t which can be known mathematically.

Here \(\frac{d^{2} x}{d t^{2}}\) double differentiation of x w.r.t. t which can he calculated mathematically.

Therefore, instantaneous acceleration is the differentiation of velocity with respect to time and is double differentiation of displacement w.r.t. time.

For any moving object at definite time intervals, if the change in velocity is also the same then this is known as uniform acceleration. And if the changes are different in velocity then this is non-uniform acceleration. In the same accelerated motion, average acceleration and instantaneous acceleration are the same.

If in any circular motion (path), the magnitude of the velocity of the moving object does not change but the direction of the moving object changes continuously, they this type of motion is also called accelerated motion.

Acceleration can be positive, negative or zero. If acceleration is positive its velocity increases. If acceleration is zero then the object moves with a constant speed (velocity). And if the acceleration is negative then the velocity of the object decreases. Hence, negative acceleration is called retardation.

Physics Notes

Speed and Velocity Physics Notes

Speed and Velocity:
Speed: The speed of something is the rate at which it moves or travels. Speed is defined as the rate of movement of a body expressed either as the distance traveled divided by the time taken or the rate of change of position with respect to time at a particular point. It is a scalar quantity that refers to “how fast an object is moving.” “The time rate of distance is called speed.
Speed = \(\frac{\text { distance }}{\text { time }}\)
The unit of speed is m/s. The dimensional formula of speed is [M⁰LT^-1]

Velocity: Velocity is the vector quantity that refers to “the rate at which an object changes its position.” Imagine a person moving rapidly one step forward and one step back-always returning to the original starting position. While this might result in a frenzy of activity. It would result in zero velocity. Because the person always returns to the original position.

The time rate of displacement is called velocity
velocity = displacement time

The following points may be noted about the speed of an object moving along a straight line:

• Speed is a scalar quantity. The magnitude of the velocity of the object is called its speed.
• The speed of an object in a particular direction is called the velocity of the object.
• The speed of an object has the same unit as that of velocity.

Types of Speed:
(a) Uniform speed: If a body covers equal distance in equal time intervals, howsoever small these intervals of time maybe is called uniform speed.

(b) Variable speed: If a body covers equal distances in unequal intervals of time or unequal distances in equal intervals of time, howsoever small these intervals of time maybe is called variable speed.

(c) Average speed: Average speed of a body is defined as the total distance travelled divided by the total time taken.
v_av = \(\frac{\Delta x}{\Delta t}=\frac{x_{2}-x_{1}}{t_{2}-t_{1}}\)

(d) Instantaneous speed:
The speed of a body at any instant of time is called instantaneous speed. When we say about speed then it means about instantaneous speed. For this time interval should be very less or Δt → 0

The instantaneous speed of the vehicle is measured by the speedometer.

Types of velocity:
(a) Uniform velocity: It is defined as the ratio of the displacement to the time taken by the object to cover the displacement.
Uniform velocity = \(\frac{\text { displacement }}{\text { timeinterval }}\)

(b) Variable velocity: A body is said to be in variable velocity if it covers equal displacement in unequal intervals of time.

(c) Average velocity: Average velocity of a body is defined as the change in position or displacement (Δx) divided by time interval (Δt) in which that displacement occurs.
\(\overrightarrow{v_{a v}}=\frac{\Delta \vec{x}}{\Delta t}=\frac{\overrightarrow{x_{2}}-\overrightarrow{x_{1}}}{t_{2}-t_{1}}\)

(d) Instantaneous velocity: The instantaneous velocity of a body is the velocity of the body at any instant of time or at any point of its path.

\(\vec{v}=\frac{d \vec{x}}{d t}\)
Velocity can be positive, negative or zero.

By studying speed and velocity we come to the result that at any time interval average speed of an object is equal to or more than the average velocity but instantaneous speed is equal to instantaneous velocity.

Physics Notes