Question

(a) Derive the formula: s = ut + 1/2 at², where the symbols have usual meanings.

Answer 1

Let the initial velocity of a body is 'u' and the final velocity of the body is 'v'.

The average velocity of the body = $\frac{u+v}{2}$

If the body takes time 't' to travel a distance of 's', then,

$s=\frac{u+v}{2}\times t$ .......(1)

Using the first equation of motion we can write, v = u +at .......(2) (Where 'a' is the acceleration of the body)

Putting the value of 'v' from equation (2) into equation (1), we get,

$$\begin{gathered} \Rightarrow s=\left(\frac{u+(u+at)}{2}\right)\times t \\ \Rightarrow s=\left(\frac{2u+at}{2}\right)\times t=ut+\frac{1}{2}a{t}^2 \\ \Rightarrow s=ut+\frac{1}{2}a{t}^2 \end{gathered}$$