User:Inconspicuum/Physics (A Level)/Kinematics

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. $$v = u + at\,$$

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

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

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

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:

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

$$at = v - u\,$$

$$v = u + at\,$$ (1)

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

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

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

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

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

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

$$at = v - u\,$$

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

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

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

$$2as = v^2 - u^2\,$$

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