At time t, the velocity v(t) of an object moving in a straight line satisfies (a)...

At time t, the velocity v(t) of an object moving in a straight line satisfies

(a) Show that , where v₀ denotes the velocity of the object at time t = 0 (and we assume v₀ > 0). Hence prove that the object comes to rest after a finite time tan⁻¹(v₀). Does the object remain at rest?

(b) Use the chain rule to show that (1.4.17) can be written as where x(t) denotes the distance traveled by the object at time t , from its position at t = 0. Determine the distance traveled by the object when it first comes to rest.