Kinematics Physics Notes

Generally we have seen different types of objects in our nearby environment in which we view the differences in their condition, of state, shape, position, colour, etc. A similar difference is viewed in their positions due to which some objects are stable (motionless) to us which means the body (object) does not change its positions as time passes. Whereas some objects change their positions with time.

For example, if we see the table in our class it seems stable (rest) to us; while a car moving outside the classroom seems as unstable (motion). Here the table does not change its position with respect to time where as the car moving on the road changes its position with respect to time. Therefore, in the physical world, we define speed in terms of the viewed change in the position of the object as time passes; which means:

“Speed is the change in the position of the atom (particle) or the collection of atoms (particles) as time passes.”

Movement of vehicle, speed of gaseous atoms, sound reaching from one place to another, speed of the cricket ball, flying of the birds and an aeroplane in the sky, speed of different astronomical or celestials bodies, blood circulation in the views and artery, breathing, etc., all such events and their occurences are examples of speed in the visible or invisible form.

Speed can be in the form of a simple straight line, in a plane, curved, circular or any other path, rotation, shiver or in the form of waves or can be in the independent form in the sky.

The branch of physics which deals with the study of motion of material object is called Mechanics. Mechanics can be classified into the following branches:
1. Statics: It is a branch of Mechanics which deals with the study of the material object at rest. An object can be at rest, even when a number of forces acting on it are in equilibrium. Thus, Statics is the study of the motion of an object under the effect of forces in equilibrium. Here, time factor does not play any role.

2. Kinematics: It is the branch of Mechanics which deals with the study of the motion of the object without taking into account the cause of their motion.
Here time factor plays an essential role. The term Kinematics is derived from the Greek word ‘Kinema’ meaning motion.

3. Dynamics: The study of the motion of the objects by taking into account the cause (or causes) of their change of state (rest or uniform motion) is called dynamics.

Concept of a point object: An object is considered a point object if the size of the object is much smaller than the distance it moves in a duration of time.
For example; the length of a train in comparison to the distance covered by the train is very less; therefore the train is assumed as a point object or as an atom (particle).

→ Motion: If an object changes its position with respect to its surroundings with time, then it is called in motion.

→ Rest: If an object does not change its position with respect to its surroundings with time, then it is called at rest.

→ Rest and motion are relative states. It means an object which is at rest in one frame of reference can be is motion in another frame of reference at the same time.

→ Point mass object: An object can be considered as a point mass object, if the distance travelled by it in motion is very large in comparison to its dimensions.

→ One dimensional motion: If only one of the three coordinates specifying the positions of the object changes with respect to time, then the motion is called one-dimensional motion. For instance, motion of a block is a straight line, motion of a train along a straight track, a man walking on a level or a and narrow road and object falling under gravity, etc.

→ Two-dimensional motion: If only two out of the three coordinates specifying the position of the object changes with respect to time, then the motion is called two-dimensional motion. A circular motion is a two-dimensional motion.

→ Three-dimensional motion: If all the three coordinates specifying the position of the object changes with time, then the motion is called 3-D motion. Flying kite, a flying airplane, the random motion of a gas molecule, etc. are examples of 3-dimensional motion.

→ Distance: The length of the actual path traversed by an object is called the distance. It is a scalar quantity and it can never be zero or negative. Its unit is metre.

→ Displacement: The shortest distance between the initial and final position of any object during motion is called displacement. It can be positive, zero or negative. It is a vector quantity. Its unit is metre.

→ When the motion of an object is along a straight line, there are only two directions i.e., forward and backward or upward and downward in which an object can move. Therefore, it is easier to represent the displacement in these two directions by sign and we do not need to use the vector notation.

→ Speed: Speed is the distance travelled per unit of time. It is a scalar quantity.

→ Velocity: Velocity is a vector representation of the displacement that an object or particle undergoes with respect to time.

→ Uniform motion: The motion of an object is said to be uniform if it covers equal displacement (or distance) in equal intervals of time. It is a vector quantity.

→ Uniform speed: It is defined as the ratio of the path length (distance) to the time taken by the object to cover the path.
Mathematically: Speed of the uniform motion = \(\frac{\text { Path length }}{\text { Timeinterval }}\)

→ Average velocity: The ratio of the total displacement to the total time taken is called average velocity.
Average velocity = \(\frac{\text { Totaldisplacement }}{\text { Total time taken }}\)

→ Acceleration: Acceleration is the rate of change of velocity of an object with respect to time.
Acceleration (a) = \(\frac{\text { Change in velocity }}{\text { Time interval }}=\frac{\Delta v}{\Delta t}\)

Its unit is m/s and the dimensional formula is [M⁰L¹T^-2]. It is a vector quantity.

→ Acceleration can be positive, zero or negative. Positive acceleration means velocity is increasing with time, zero acceleration means velocity is uniform while negative acceleration (retardation) means velocity is decreasing with time.

→ Uniformly accelerated motion: The motion of an object is said to be uniformly accelerated. If the same change in its velocity takes place in each unit of time.

→ Position time graph: It is a straight line inclined to the time-axis and the velocity of the uniform motion is equal to the slope of the position-time graph.

→ Velocity-time graph: It is a straight line inclined to the time axis.
(a) The acceleration of uniformly accelerated motion is equal to the slope of the velocity-time graph.
(b) The area under the velocity-time graph between the instant and t₂ is equal to the displacement of the object in the time interval (t₂ – t₁)

→ Equations of uniformly accelerated motion:
If a body starts with velocity (u) and after time t its velocity changes to v, if the uniform acceleration is ‘a’ and the distance travelled in time t is ‘ s’, then the following equations are called equations of motion.

v = u + at
s = ut + \(\frac{1}{2}\) at²
v² = u² + 2as
Distance travelled in nth second
S_n = u + \(\frac{a}{2}\)(2n – 1)

→ Motion under effect of gravity: If an object is falling freely (u = 0) under gravity, then the equations of motion are;

v = u + gt
h = ut + \(\frac{1}{2}\)gt² and
v² = u² + 2gh

Note: If an object is thrown upward then ‘g is replaced by -g in the above three equations.

→ Projectile motion: When any object is thrown from a horizontal angle θ, except 90°, then the path followed by it, is called trajectory, the object is called projectile and its motion is called projectile motion.

If any object is thrown with velocity u, making an angle θ, from horizontal, then
(a) Equation of the path of a projectile
y = xtanθ – \(\frac{g}{2 u^{2} \cos ^{2} \theta}\) x²
The path of a projectile is parabolic.

(b) Time of flight: It is defined as the total time for which the projectile remains in the air.
T = \(\frac{2 u \sin \theta}{g}\)

(c) Maximum height: It is defined as the maximum vertical distance covered by the projectile.
H = \(\frac{u^{2} \sin ^{2} \theta}{2 g}\)

(d) Horizontal range: It is defined as the maximum distance covered by the object in horizontal direction.
R = \(\frac{u^{2} \sin 2 \theta}{g}\)

(e) Horizontal range is maximum when it is thrown at angle of 45° with the horizontal
Rmax = \(\frac{u^{2}}{g}\)

(f) For angle of projection (θ) and (90-θ) the horizontal range are same.

→ Relative-velocity: The relative velocity of an object w.r.t. another moving object is the effective velocity with which the object will appear to move when the other object is considered to be at rest.

Mathematically: If VA and VB are the velocities of the two objects A and B, then the relative velocity of an object A w.r.t. the object B is given by
V_AB = V_A + (-V_B)
(a) When the two objects are moving along a straight line in the same direction, the relative velocity is
V_AB = V_A – V_B
(b) When the two objects are moving along a straight line in the opposite directions, the relative velocity is
V_AB = V_A + V_B

→ The frame of reference: A frame of reference is a set of coordinates that can be used to determine the positions and velocities of objects in that frame.

→ Translatory motion: Translatory motion is the motion by which. a body shifts from point to another w.r.t. the coordinate system. for example motion of a vehicle on straight road, horizontal motion of a body, etc.

→ Rotational motion: When a body constantly moves around an fixed axis it is called a rotational motion for example motion of ceiling fan, a string whirled in a circular loop, etc.

→ Oscillatory vibrational motion: The motion in which a particle moves to and fro motion about a given point is called oscillatory or vibrational motion, for example motion of a simple pendulum, motion of a mass attached to a spring, etc.

→ Retardation/Deceleration: Negative acceleration is called. Retardation or deceleration.

→ Uniform accelerated motion: In uniform acceleration, the magnitude and direction of an object always remain constant. This type of motion is called uniform accelerated motion.

→ Relative velocity: 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.