Block pushed by time-varying force: momentum, speed, and power

[Tanero]

Homework Statement

A block of mass m=2 kg is initially at rest on a horizontal surface. At time t=0, we begin pushing on it with a horizontal force that varies with time as F(t)=Beta t^2, where Beta = 1.0 N/s^2. We stop pushing at time t1, 5s [F(t)=0 for t>1].
(a) First, assume the surface is frictionless. What is the magnitude of the final momentum at t1=5s?
P final(t=t1)=
(b) let's now consider a new situation where the object is initially at rest on a rough surface. The coefficient of static friction is 0.2. What is the speed of the block at time t2=5s?
v(t=t2)=
(c) What is the power P provided by the force F(t) at time t3=4s (in Watts) i the case where there is friction (part(b))?
P(t=t3)=

Homework Equations

Can you please check if my approach is right? I have some doubts.
p=∫Fdt ,
instantaneous power P=Fv

The Attempt at a Solution

part a) p=∫βt²=1/3βt³

part b) F - f_k = ma
F - f_k = mdv/dt
(F - f_k)dt = mdv
∫(F - f_k)dt = ∫mdv
mv_f - mv_i = (β/3)(t_f)^3 - (f_k)t_f - [(β/3)(t_i)^3 - (f_k)t_i] [v_i = 0 and t_i = 0]
mv_f = (β/3)(t_f)^3 - (f_k)t_f

part c)
P=Fv
to find v I used same approach as for b (but for t=4) and F=1*4²

 

[TSny]

Your work looks good. You didn't include β in your expression for the answer in (a), but I guess that's because it has a numerical value of 1 N/s².

Of course you will need to plug in the time values and figure out the friction force to get your final numerical answers.

Good work.

[Welcome to PF!]