Question

A retarding force f=-2v is acting on a body of mass 10 gram. Find out time taken for its velocity to reduce to 37% of its initial value if initial velocity is 1m/s

Answer 1

According to the Newton's second law and according to the task

$$m\frac{dv(t)}{dt}=-2v(t)$$

Solution of the equation with the initial condition

$$v(0)=v_0$$

is

$$v(t)=v_0\exp \left(-\frac{2t}{m}\right)$$

So the time taken for its velocity to reduce to 37% of initial value can be found from the equation

$$v(t)=0.37v_0$$

The solution is

$$t=-\frac{m}{2}\ln(0.37)$$

And according to the task

$$t\approx5\:\mathrm{ms}$$

We can see that the result doesn't depend on initial velocity.