Kinematics is the study of how objects move. One needs to understand a situation in which an object changes speed, accelerating or decelerating, and travelling a certain distance. There are four equations you need to be able to use which relate these quantities.

## Variables

Before we can understand the kinematic equations, we need to understand the variables involved. They are as follows:

• t is the length of the interval of time being considered, in seconds.
• v is the speed of the object at the end of the time interval, in ms−1.
• u is the speed of the object at the beginning of the time interval, in ms−1.
• a is the acceleration of the object during the time interval, in ms−2. Has to be a constant.
• s is the displacement (distance traveled) of the object during the time interval, in meters.

## Equations

The four equations are as follows:

1. ${\displaystyle v=u+at\,}$

2. ${\displaystyle s={\frac {u+v}{2}}t}$

3. ${\displaystyle s=ut+{\frac {at^{2}}{2}}}$

4. ${\displaystyle v^{2}=u^{2}+2as\,}$

## Derivations

It is also useful to know where the above equations come from. We know that acceleration is equal to change in speed per. unit time, so:

${\displaystyle a={\frac {v-u}{t}}}$ (*)

${\displaystyle at=v-u\,}$

${\displaystyle v=u+at\,}$ (1)

We also know that the average speed over the time interval is equal to displacement per. unit time, so:

${\displaystyle {\frac {u+v}{2}}={\frac {s}{t}}}$

${\displaystyle s={\frac {u+v}{2}}t}$ (2)

If we substitute the value of v from equation 1 into equation 2, we get:

${\displaystyle s={\frac {u+(u+at)}{2}}t={\frac {2u+at}{2}}t=t(u+{\frac {at}{2}})=ut+{\frac {at^{2}}{2}}}$ (3)

If we take the equation for acceleration (*), we can rearrange it to get:

${\displaystyle at=v-u\,}$

${\displaystyle t={\frac {v-u}{a}}}$

If we substitute this equation for t into equation 2, we obtain:

${\displaystyle s={\frac {u+v}{2}}{\frac {v-u}{a}}={\frac {(v+u)(v-u)}{2a}}={\frac {v^{2}-u^{2}}{2a}}}$

${\displaystyle 2as=v^{2}-u^{2}\,}$

${\displaystyle v^{2}=u^{2}+2as\,}$ (4)

## Questions

1. A person accelerates from a speed of 1 ms−1 to 1.7 ms−1 in 25 seconds. How far has he travelled in this time?

2. A car accelerates at a rate of 18 kmh−2 to a speed of 60 kmh−1, travelling 1 km in the process. How fast was the car travelling before it travelled this distance?

3. A goose in flight is travelling at 4 ms−1. It accelerates at a rate of 1.5 ms−2 for 7 seconds. What is its new speed?

4. How far does an aeroplane travel if it accelerates from 400 kmh−1 at a rate of 40 kmh−2 for 1 hour?