Problem 664 sace trig t onta FoetT is exerted on the object

Problem 6.64 sace trig t onta Foe-t/T is exerted on the object. At t 0, an object of mass m is at rest at z = 0 on a horizontal, frictionless surface. Starting at t = 0, a horizontal force

Solution

The force acting on the particle is given as Foe^-t/T

We know that acceleration = Force / mass

Hence, acceleration = a(t) = (Fo/m)e^-t/T

Further, for any motion, we know that dv /dt = a(t)

Rearrganging the terms, we get:

dv = a(t) dt

or dv = (Fo/m)e^-t/T dt

We will have to integrate the above relation to get the velocity at any time t

Part A: Hence, V = -(FoT/m)[e^-t/T - 1] = (FoT/m)[1 - e^-t/T]

Part B:

For the above relation of V, we need to find the value of V for t tending to infinity. For t tending to infinity, e^-t/T will tend to zero.

Hence, Velocity after a very long time = (FoT/m)

NOTE: For questions involving force or acceleration dependent on time, we would need to employ dv/dt = a(t) to determine the relation for dV which can then be integrated to obtain V at any point. Plus, similar relation as

ds/dt = v(t) can be used in case displacement is to be determined.