I'm not sure if my approach to this question is correct. I also don't understand why I'm getting inconsistent results from my own working.
Question
Working
Part 1:
The time that the block moves is the moment where the push force overpowers the force due to friction
Then we have
$$ F_{push} = F_{friction} $$
Our values are
$$ F_p = 3t, F_f = \mu_s m g = \mu_s g = 10(0.6) = 6 $$
From this we have that the block will move at the moment
$$ 3t = 6 $$
Solving for $t$ gives
$$ t = \frac{6}{3} = 2 $$
Then for the first part we have that the block starts to move at time $t = 2$ seconds.
Part 2 (using Newtons Second):
We want to know the speed of the block at $t = 5$ seconds.
From part 1 we know that the block doesn't start to move until time $ t = 2$ seconds.
From newtons second law we have
$$ F = ma $$
Where $F$ is the net force. This net force will be $F_n = F_p - F_f$ (push - friction).
$$ F_p = 3t $$
$$ F_f = \mu_k m g = \mu_k g = 0.55(10) = 5.5 $$
Then we have
$$ F_n = 3t - 5.5 = ma = a $$
So this is our acceleration function, integration of this gives us velocity as
$$ v = \frac{3}{2}t^2 - 5.5t $$
The constant of integration is dropped as initial velocity is zero.
Therefore at time $t = 5$ seconds we have speed
$$ v = \frac{3}{2}(5^2) - 5.5(5) = 10 $$
this shows that the speed is 10 m/s at time $ t = 5$ seconds
Part 2 (using conservation of momentum):
Conservation of momentum states that
$$ p_{init} = p_{final} $$
We also have Impulse as
$$ I = F \Delta t $$
Here $\Delta t = 3$ as we're moving from $t = 2$ to $t = 5$ seconds. We also have the net force as before, which is $F = ma = a = 3t - 5.5$
Which puts Impulse as
$$ I = (3t - 5.5) \times 3 = 9t - 16.5 $$
Using $I = \Delta p$ where $\Delta p = m(v_f - v_0)$, note that as $m = 1$ we have $\Delta p = (v_f - v_0)$, and here $v_0 = 0$, so we just have $\Delta p = v_f$.
Using this gives
$$ I = 9t - 16.5 = v_f $$
Inputting $t = 5$ gives
$$ v_f = 45 - 16.5 = 28.5 $$
So
So for part 1 I got $t = 2$ seconds
For Part 2 (using Newtons) I got $v = 10$
For Part 2 (using momentum) I got $v = 28.5$
So clearly something here is wrong as I have inconsistent results, I'm not sure what though.