Solving for Space Vehicle Acceleration: Homework Statement

[Abid Rizvi]

Homework Statement

The first stage of a space vehicle consumed fuel and oxidizer at the rate of 1.60

104 kg/s with an exhaust speed of 3.05

103 m/s. Find the acceleration the vehicle had just as it lifted off the launch pad on the Earth, taking the vehicle's initial mass as 3.00

106 kg.

Homework Equations

Momentum = mv

The Attempt at a Solution

So I tried to use math logic, but I'm not sure if its even logical...
Mi = initial mass of rocket
Vi = initial velocity of rocket (0)
Ve = velocity of exhaust stuff
dV = change in velocity
dM = change in mass of rocket
dm = change in mass of stuff expelled
dM = - dm

(Mi+dM)(Vi+dV) + Vedm = 0
(Mi+dM)(Vi+dV) - VedM = 0
MiVi + VidM +MidV+dMdV - VedM
Vi = 0, so
MidV+dMdV = VedM
multiply everything by (1/dt):
Mi*dv/dt + dM*dV/dt = Ve*dM/dt
solving for since acceleration = dv/dt, solving for dv/dt gets:
a = (Ve*dM/dt)/(Mi+dM) = -16.4.

It cannot be a negative acceleration so obviously something is wrong...

 

[SteamKing]

Homework Statement

The first stage of a space vehicle consumed fuel and oxidizer at the rate of 1.60

104 kg/s with an exhaust speed of 3.05

103 m/s. Find the acceleration the vehicle had just as it lifted off the launch pad on the Earth, taking the vehicle's initial mass as 3.00

106 kg.

Homework Equations

Momentum = mv

The Attempt at a Solution

So I tried to use math logic, but I'm not sure if its even logical...
Mi = initial mass of rocket
Vi = initial velocity of rocket (0)
Ve = velocity of exhaust stuff
dV = change in velocity
dM = change in mass of rocket
dm = change in mass of stuff expelled
dM = - dm

(Mi+dM)(Vi+dV) + Vedm = 0
(Mi+dM)(Vi+dV) - VedM = 0
MiVi + VidM +MidV+dMdV - VedM
Vi = 0, so
MidV+dMdV = VedM
multiply everything by (1/dt):
Mi*dv/dt + dM*dV/dt = Ve*dM/dt
solving for since acceleration = dv/dt, solving for dv/dt gets:
a = (Ve*dM/dt)/(Mi+dM) = -16.4.

It cannot be a negative acceleration so obviously something is wrong...

IDK why you are using 'math logic', whatever that is. Rocket science is mostly about physics, so you should have your physics cap on.

If you start with first principles, F = ma always keeps popping up, but this form of Newton's second law isn't in its most general form. What you want is to write the second law in terms of a change in momentum for the rocket and the exhaust products, like so:

where Fi is the net force acting on the rocket, m is the mass of the rocket at time t, v is the velocity of the rocket, ve is the velocity of the exhaust, and dm is the change in the mass of the rocket due to burning the rocket fuel.

For more details, see:

http://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation

 

[Orodruin]

All of the quantities in your expression are positive in the way you have defined them so you should find a positive answer. You also should put some units on your answer or it will be unclear what you are actually answering (i.e., in this case m/s^2).

Another issue is: Should you include gravity in the acceleration of the rocket? If it is just taking off from the launch pad, it is definitely not negligible.