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Dynamics: Kinematics

Kinematics

Before going further, we discussed that there are two types of branches where we deal with motion in two different ways. One was considering the force which cause the motion and second, the one without considering the cause of motion. The study of motion without considering the cause of it is known as Kinematics.

This can be easily explained with the fact we just consider objects velocities, acceleration, time period and displacement which are commonly known by “suvat” i.e.

s = displacement

u = initial velocity

v = final velocity

a = uniform acceleration

t = time period

Before going any further on the kinematics, there are certain things that we need to understand, which are as mentioned below:

  1)    Rest: An object is said to be at rest, if its position doesn’t change with respect to surrounding. For example: Houses are at rest from our perspective if we are not moving.

  2)    Motion: An object is said to be in motion, if its position changes with respect to surrounding. For example: A moving bus is said to be in motion with reference to us if we are at rest.

  3)    Distance: It is the actual length of path travelled by an object between its initial and final position.

  4)    Displacement: It is the shortest length of path between the initial and final position.

  5)    Speed: The rate of change of distance with respect to time is speed. Mathematically, it is defined as the modulus (magnitude) of velocity.

  6)    Velocity: The rate of change of displacement with respect to time is velocity.

  7)    Acceleration: The rate of change of velocity with respect to time is acceleration.


Types of motion:

Considering the types of motion, there are two types of motion; either the object is moving with uniform velocity or its velocity is changing instantly.

Simple Examples of the body moving with uniform velocity and changing velocity: 

Equations for Uniformly Accelerated motion:

Let us prove these with the help of velocity time graph, which is mentioned below: Consider s, u, v, a, t be the displacement, initial velocity, final velocity, uniform acceleration and time taken.

From the above diagram, we have

NQ = v - u

PQ = t

Then we know

Slope of velocity time graph is equal to the acceleration:

So,


Also, we know that,

Area of the velocity time graph gives the displacement, so

 This way we can prove the equations of motion through the velocity time graph.



Distance covered in the tth  second:

 The distance covered in the tth second is the difference between the distance travelled in the t second and distance travelled in the (t-1) second. So,

Motion Under Gravity:

Here, the body is dropped from the top of the tower of height h and body falls completely under the influence of gravity. So, there will be just modifications in our formulas mentioned as the motion will indeed be in the straight line and directions are taken positive or negative according to the direction. i.e.

Motion in an inclined plane:

The motion of a body in an inclined plane will be linear as the object will slide down or slide up along the length of the inclined plane.

 The general figure of object on the inclined plane is mentioned below:

The components of the g will be decomposed along the plane and perpendicular to the plane. The component along the plane will be the acceleration of the body on the plane if we consider the plane to be smooth.

 The motion of the body will be in a straight line hence the formulas mentioned for general motion in the straight line can be used with some modifications. If l is the length of the inclined plane and theta is the angle of inclined plane with respect to horizontal then,

For the sign, we basically use conventions. If the body’s velocity acceleration and displacement are taken in same direction, everything will be positive and if any one of them is in opposite direction we take that as negative.