What is the acceleration of the rocket?

[kalupahana]

1. Problem
A launched rocket emitting gas with a speed of 1500 m/sec relative to rocket and with flowrate of 100 kg/sec.
If net mass of rocket is 5000 kg, find it's acceleration.

2. Equations
F=ma
F=d(mv)/dt
Conservation of momentum
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

3.Attempt
Flowrate = 100 kg/s
velocity = 1500 m/s
momentum downwards = mv = 100 x 1500 = 1.5 x 105 kg m/s

According to conservation of momentum, i found the velocity to upward direction.

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
0 = -100 x 1500 + 5000v₂
5v₂ = 150
v₂ = 30 m/s

what should do next to find the acceleration
If I use F=d(mv)/dt
what is the initial velocity and final velocity. I have no idea about those two & and how to take it.

 

[housemartin]

Hello!
usually those kind of problems are done first writing momentum at the time t, then at the time t+dt, taking difference P(t+dt)-P(t) and dividing it by dt, then you will get Newton's second law: F= dP/dt, with no external forces F = 0. If you'll get correct expression for P(t) and P(t+dt), then acceleration will pop up in dP/dt expression. just don't forget that mass is changing, and so is velocity, and that gas emitting speed is given relative to the rocket - in writing momentum you need its speed relative to inertial frame (one that is not accelerating).