Answer:
6900 m/s
Explanation:
The mass of the rocket is:
m = 330000 − 280000 (t / 250)
m = 330000 − 1120 t
Force is mass times acceleration:
F = ma
a = F / m
a = F / (330000 − 1120 t)
Acceleration is the derivative of velocity:
dv/dt = F / (330000 − 1120 t)
dv = F dt / (330000 − 1120 t)
Multiply both sides by -1120:
-1120 dv = -1120 F dt / (330000 − 1120 t)
Integrate both sides. Assuming the rocket starts at rest:
-1120 (v − 0) = F [ ln(330000 − 1120 t) − ln(330000 − 0) ]
-1120 v = F [ ln(330000 − 1120 t) − ln(330000) ]
1120 v = F [ ln(330000) − ln(330000 − 1120 t) ]
1120 v = F ln(330000 / (330000 − 1120 t))
v = (F / 1120) ln(330000 / (330000 − 1120 t))
Given t = 250 s and F = 4.1×10⁶ N:
v = (4.1×10⁶ / 1120) ln(330000 / (330000 − 1120×250))
v = 6900 m/s
