Question

if initial speed of particle is u and its accelerationis given as a=At^3 where A is constant and t is time, then its final speed v will be given as
1) u+At^4
2) u+At^4/4
3) u+At^3
4) u+At^3/3​

Answer 1

Given that,

Initial Velocity of the particle is u and acceleration is At$^3$

Since, acceleration varies w.r.t time.

$\longrightarrow \sf \: \dfrac{dv}{dt} = A {t}^{3} \\ \\ \longrightarrow \sf dv = A {t}^{3} dt$

Integrating w.r.t time on both sides,

$\longrightarrow \displaystyle \int\sf dv = \int A {t}^{3} dt \\ \\ \longrightarrow \displaystyle \int\sf dv = A \int {t}^{3} dt \\ \\ \longrightarrow \displaystyle \sf v \bigg | {}^{v} _u = A \dfrac{ {t}^{4} }{4} \bigg| {}^{t} _0 \\ \\ \longrightarrow \sf \: v - u = \dfrac{A}{4} ( {t}^{4} - 0 {}^{4} ) \\ \\ \longrightarrow \boxed{ \boxed{\sf \: v = u + A \dfrac{{t}^{4} }{4} }}$

Option (2) is correct