Question

An external force acting on a particle of mass m in one dimension is

given by F = βt2 where β is a positive constant. If the initial speed of the particle is

v1 (at t = 0) and its final speed is v2 at t = tf . Find constant β in terms of the given

quantities.

Answer 1

Answer

Force

$F=m\frac{dv}{dt}={\beta t^2}$

$=\frac{dv}{dt}=\frac{\beta t^2}{m}$

By integration v limit v1 to v2 and time t =0 to t=t_f

So we get

$v_2-v_1=\frac{\beta t^3 }{3m}$

So constant

$\beta=\frac{(v_2-v_1) 3m}{t_f^3}$