Question

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

Answer 1

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 $\Rightarrow \dfrac{InitialVelocity+FinalVelocity}{2}$
$\Rightarrow \dfrac{u+v}{2}$
And Distance travelled=Average velocity × time
Therefore, $s\Rightarrow \left( \dfrac{u+v}{2} \right)\times t$
We know the first equation of motion, $v=u+at$
Now we are putting all the above value in equation ,–

& s\Rightarrow \left( \dfrac{u+u+at}{2} \right)\times t \\\ & s\Rightarrow \left( \dfrac{2ut+a{{t}^{2}}}{2} \right) \\\ & \\\ & \\\ \end{aligned}$$ $$s=ut+\dfrac{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.