Question

An object of mass 3kg is at rest. Now a force of F = 6t 2 i + 4t j is applied on the object then velocity of object at t = 3 second is : -

Answer 1

Force by definition is rate of change of momentum. In the present case, the force is time dependent. Let at any moment t the velocity of the body be u. Let the velocity change to v in the time interval dt ie at time t+dt the velocity of the body is v.

Change in momentum of the body in time interval dt = m v - m u.

Force on the body (assumed unchanged during the interval dt ) = (m v - m u)/dt.

Change in momentum in time interval dt = F dt= ( 6 t² i + 2 t j ) dt.

In the present case, the force is time dependent, so by integrating we can get the change in momentum and on putting the limits we get value of change in momentum during that interval. Integral yields,

Change in momentum = [ 6 t³ i /3 + 2 t² j/2]= 2 t³ i + t² j.

Putting the limits 0s to 3s, in 2 t³ i + t² j, we get the change in momentum in the body in the time interval 0s to 3s = 54 i + 9 j.

So change in momentum = 54 i + 9 j= m v - mu.

Diving by the mass 3 kg we get change in velocity in the interval 0s to 3s= 18 i + 3 j .

Now at time t= 0s, the velocity u = 0 as the body is at rest.

So velocity of the body at t=3s is 18 i + 3 j.

Due to lack of symbols for integation, and I not knowing latex I took recourse to description of the integration process, but I wrote the answer in the hope it adds to our understanding on how to solve such cases.