Question

(A) Derive the formula: $$s=ut+{\displaystyle \frac{1}{2}}a{t}^2$$, where the symbols have usual meaning.

Answer 1

Hint: In ordered to answer the first part of the question we would assume some state of a body to prove the above formula. The 1st equation of motion helps us to derive the above formula.

Complete step-by-step solution:
(A) Let us assume a body has an initial velocity = $$u$$,
Uniform acceleration = $$a$$,
Time= $$t$$,
Final velocity= $$v$$,
Distance travelled by the body= $$s$$,
Now calculate the average velocity $$\Rightarrow {\displaystyle \frac{InitialVelocity+FinalVelocity}{2}}$$
                                                                $$\Rightarrow {\displaystyle \frac{u+v}{2}}$$
And Distance travelled=Average velocity × time
Therefore, $$s\Rightarrow \left({\displaystyle \frac{u+v}{2}}\right)\times t$$
We know the first equation of motion, $$v=u+at$$
Now we are putting all the above value in equation ,–
$$\begin{array}{rl}& s\Rightarrow \left({\displaystyle \frac{u+u+at}{2}}\right)\times t\\ & s\Rightarrow \left({\displaystyle \frac{2ut+a{t}^2}{2}}\right)\\ & \\ & \end{array}$$
$$s=ut+{\displaystyle \frac{1}{2}}a{t}^2$$ where,
initial velocity = $$u$$,
Uniform acceleration = $$a$$,
Time= $$t$$,
Final velocity= $$v$$,

Note: Uniform acceleration remains constant with respect to the time . There are some examples of uniform accelerated motion –i.e. dropping a ball from the top, car going along a straight road, skydiver jumping out of the plane.