Velocity is defined as the rate of change of displacement.
$v=\frac{\Delta\ s}{\Delta\ t}$ ,
Average velocity is the ratio of total displacement to total time taken.
Average velocity =$\frac{\text{total displacement}}{\text{total time taken}}$
$\implies \frac{u+v}{2}$ = $\frac{s}{t}$
$\implies s=\bigg(\frac{u+v}{2}\bigg)\times t$
Acceleration is defined as the rate of change of velocity.
$a=\frac{\Delta\ v}{\Delta\ t}$
$a=\implies \frac{v-u}{t}$
$\implies$ v = u +at
The above two equations can be combined and the other equations of motion can be derived.
The equations of motion or Kinematic equations are listed below:
- v = u + at
- v$^{2}$= u$^{2}$+ $2$as
- s = ut + $\frac{1}{2} \text{at}^{2}$
- s = vt – $\frac{1}{2}\text{at}^{2}$
- s =$\bigg(\frac{u+v}{2}\bigg)\times$ t
  
  where,
  u = initial velocity, v = final velocity, a = acceleration, s = displacement and t = time
In the above equations, except for time, all the other quantities are vectors.
- The positive and negative directions of the quantities: velocity, acceleration and displacement should be considered beforehand.
- Suppose, the right side is considered as the positive direction, then the left side will be the negative direction as shown below:
Here’s an experiment to determine the acceleration of free fall:
- As a metal ball is released from rest from an electromagnet to a trapdoor, the total time taken for the fall is measured using the timer.
- Upon measuring the height using a metre rule, we can simply use the equation shown below to find the acceleration of free fall or the acceleration due to gravity.
  s= $\frac{1}{2}\text{gt}^{2}$

While solving the problems related to motion in a uniform gravitational field, the direction for positive and negative should be set beforehand. And, the replacement of acceleration ‘a’ with the acceleration due to gravity ‘g’ is to be done accordingly in the above 5 equations of motion.
Example:

For a body falling in a uniform gravitational field without air resistance, graphs can be used to demonstrate its movement, as seen here:
- Displacement-time graph,
- A velocity-time graph
- An acceleration-time graph

2.02 - Kinematics II

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