Question

The initial velocity of a particle is u and the acceleration is given by (kt), where k is a positive constant. The distance traveled in time t is :-

• A. $s=u{t}^2+k{t}^2$
• B. $s=ut+\left(\frac{k{t}^3}{6}\right)$
• C. $s=ut+\left(\frac{k{t}^3}{2}\right)$
• D. $s=\left(\frac{u{t}^2}{2}\right)+\left(\frac{k{t}^3}{6}\right)$

Answer 1

Answer: (B)

$s=ut+\left(\frac{k{t}^3}{6}\right)$

We have $a=kt$

Then,

Velocity, $v=\int adt=\int ktdt=k\int \frac{t^2}{2}+c$

$⟹v=\frac{k{t}^2}{2}+u$

Then,

$s=\int vdt={\int }_0^t(\frac{k{t}^2}{2}+u)dt$

$⟹s=\frac{k{t}^3}{6}+ut$