Acceleration, thrust, and propellant velocity?

[camdickman]

Homework Statement

What is the average acceleration until burnout of a spacecraft engine creating 53.2MN of thrust with a propellant velocity of 4.78km/s? The initial mass is 2.12x10^6 kg and final ,*** is 7.04x10^4.

Homework Equations

0 = m(subscript_f)v(_f) + m(_r)v(_r)

The Attempt at a Solution

0 = m(subscript_f)v(_f) + m(_r)v(_r)
0 = (2.12x10^6 - 7.04x10^4)*(4.78x10^3)
v(_r) = 139,163m/s
The problem is, I don't know where to go from here in order to determine the average velocity from here... Help!

 

[Andrew Mason]

Homework Statement

What is the average acceleration until burnout of a spacecraft engine creating 53.2MN of thrust with a propellant velocity of 4.78km/s? The initial mass is 2.12x10^6 kg and final ,*** is 7.04x10^4.

Homework Equations

0 = m(subscript_f)v(_f) + m(_r)v(_r)

The Attempt at a Solution

0 = m(subscript_f)v(_f) + m(_r)v(_r)
0 = (2.12x10^6 - 7.04x10^4)*(4.78x10^3)
v(_r) = 139,163m/s
The problem is, I don't know where to go from here in order to determine the average velocity from here... Help!

Start with:
$$a_{avg} = \frac{\Delta v}{\Delta t}$$

So you have to find the change in velocity and the time until burnout. Find the time to burnout first and then find the change in velocity.

In order to do this, you have to find the rate at which mass is being ejected: dm/dt, which is constant. Since you know the total change in mass you can easily determine the time to burnout.

The change in velocity of the rocket is a little more difficult to work out. You have to do an integration.

Hint: dp/dt = F = ma = constant = 53.2 MN

$m = m_0 - (dm/dt)t$

$\Delta v = \int_0^t dv = \int_0^t a dt$

AM