Rocket problem; impulse, change in velocity
at t=0, a 2250kg in outer space fires an engine that exerts an increasing force on it in the +x-direction. this force obeys the equation
fx=at2
where 1.25s is time, and has a magnitude of 781.25n when t=1.25s
1) find the si constant a.
2) what impulse does the engine exert on the rocket during the 1.50s interval starting 2.00s after the engine was fired?
3) by how much does the rocket's velocity change during this interval?

1) Find the SI constant A.

Fx = A(t^2)

t = 1.25 s => Fx = A(1.25^2) => A = Fx / (1.25^2) = 781.25N / (1.25^2) s^2 = 500N/s^2

A = 500 N/s^2


2) What impulse does the engine exert on the rocket during the 1.50s interval starting 2.00s after the engine was fired?

I = Integral of Fx * dt from t = 2.0 s to t = 3.5 s

I = Integral of A(t^2)dt from t = 2.0s to t = 3.5s

I = 500(t^3) / 3 from t = 2.0 s to t = 3.5s

I = [500/3]* [(3.5s)^3 - (2.0s)^3 ] = 5812.5 N*s

3) By how much does the rocket's velocity change during this interval?

I =  momentum change ; constant mass => m* velocity change =>

velocity change = I / m = 5812.5 N*s / 2250kg = 2.58 m/s