The Rocket Equation

Goal:

The purpose of this simulation is to give users insight into the motion of a rocket and the "Rocket Equation" that is derived on page 137 of K&K. Users will view the motion of an accelerating rocket (that is not under the influence of gravity) and the velocity (red) and position (green) as functions of time.

Basic Premise:

A rocket is cruising along at a constant speed when the pilot gets a distress call, requesting her to return as quickly as possible to her home planet. The pilot decides to burn off her 3 kg of fuel (at a constant rate) to reach a maximum speed. Watch the motion and velocity of the rocket, but pay attention to the fuel mass as well.

Analysis:

We know that the rocket equation says

F = M(dv/dt) - u(dm/dt)

We have a rocket that has no external forces on it, and we want to find its velocity after it burns off the fuel. We have

M(dv/dt) = u(dm/dt)

We can cancel out the dt’s and, after separating variables, take the integral of both sides:

∫ dv = ∫ (u/M) dm

We want to know how the velocity changed after releasing the gas, and so that’s why the limits of the left side are from the initial velocity to the final velocity, and for the right side they are from the initial mass to the final mass. Solving, we get

v_f - v_o = u*ln(M_f/M_o)

where v_f = final velocity, v_o = initial velocity, M_f = final mass, and M_o = initial mass. This is the solution to the Rocket Equation when no external force is applied to the system. It’s different if there is an external force, of course, but now that you know the steps to solve it, you can do it yourself :) (see K+K page 139 if you want an example).

The rocket attains its greatest speed when all the fuel mass is burned off, and then the acceleration stops because the total mass of the rocket no longer changes. That is why once the fuel mass = 0, we see the velocity plot become a horizontal line; the rocket reached a constant maximum velocity. Then, the position became a straight line because the velocity was constant.