Question

A body of mass m rests on a horizontal plane with the friction coefficient k. At the moment `t=0` a horizontal force is applied to it, which varies with time as `F=at`, where a is a constant vector.
Find the distance traversed by the body during the first t seconds after the force action began.

Answer 1

Since, the applied force is proportional to the time and the frictional force also exists, the motion does not start just after applying the force. The body starts its motion when F equals the limiting friction.
Let the motion start after time `t_0`, then
`F=at_0=kmg` or, `t_0=(kmg)/(a)`
So, for `t=let_0`, the body remains at rest and for `tgtt_0` obviously
`(mdv)/(dt)=a(t-t_0)` or, `mdv=a(t-t_0)dt`
Integrating, and noting `v=0` at `t=t_0`, we have for `tgtt_0`
`underset(0)overset(v)intmdv=aunderset(t_0)overset(t)int(t-t_0)dt` or `v=(a)/(2m)(t-t_0)^2`
Thus `s=int vdt=(a)/(2m)underset(t_0)overset(t)int (t-t_0)^2dt=(a)/(6m)(t-t_0)^3`