Question

Having gone through a plank of thickness h, a bullet changed its velocity from `v_0` to `v`. Find the time of motion of the bullet in the plank, assuming the resistance force to be proportional to the square of the velocity.

Answer 1

According to the problem
`m(dv)/(dt)=-kv^2`or, `m(dv)/(v^2)=-kdt`
Integrating, within the limits,
`underset(v_0)overset(v)int(dv)/(v^2)=-k/m underset(0)overset(t)intdt` or, `t=m/k((v_0-v))/(v_0v)` (1)
To find the value of k, rewrite
`mv(dv)/(ds)=-kv^2`, or, `(dv)/(v)=-k/mds`
On integrating
`underset(v_0)overset(v)int(dv)/(v)=-k/m underset(0)overset(h)intds`
So, `k=m/h1n(v_0)/(v)` (2)
Putting the value of k from (2) and (1), we get
`t=(h(v_0-v))/(v_0v1nv_0/v)`