Launching a Rocket

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

The launch of a spacecraft is fundamental to all space activity. As a rocket flies it loses mass, because most of its mass is fuel (pure hydrogen and oxygen) that provides the propulsive force.


This Demonstration shows the dynamics of an ideal rocket from launch time to measured burn-out time, based on Newton's laws and Tsiolkovsky's rocket equation. Specify the rocket parameters and launch with the trigger.


Contributed by: Frederick Wu (October 2008)
Open content licensed under CC BY-NC-SA



Graphic, top right: Tsiolkovsky's rocket equation, which defines the relationship between exhaust velocity and mass ratio

Graphic, bottom right: rocket dynamic parameters of acceleration, velocity, altitude and burn-out time

Graphic, left: 3D rocket dynamic launch model with altitude function control

Snapshot 1: ideal rocket specification that surpasses first escape velocity

Snapshot 2: ideal rocket specification that surpasses second escape velocity

Snapshot 3: 3D Earth scale model, with the launch site (yellow) and rocket altitude (red)

simplified assumptions:

1. gravity and aerodynamic drag effects are neglected

2. single stage rocket, initial velocity is zero

3. vertical launch or pitch angle is 90°

4. a constant exhaust velocity

5. a constant mass flow rate

governing equations:

1. Tsiolkovsky's rocket equation or rocket velocity:

2. mass flow rate:

3. maximum flight time or fuel burn‐out time:

4. rocket altitude:

5. rocket acceleration: , from and above governing equations 1 and 2


is the rocket velocity.

is the efficient exhaust velocity, constrained within a range 2500-4500 , using today's liquid‐fueled rocket chemical technology.

is the initial rocket mass.

is the current or final rocket mass; decreases during flight until all the liquid fuel is burned out.

is the final rocket mass, usually regarded as payload.

is the mass ratio, , which is usually in the range from 3 to 8; 14 is difficult to achieve.

is the mass flow rate, which depends on rocket engine design and specification; it indicates the rate at which the mass of the rocket is decreasing. Also called "specific impulse".

is the rocket flight time, , which is the fuel burn-out time or the maximum flight time.

is the rocket acceleration. It is difficult for the human body to withstand high acceleration; 15-20 G is the maximum tolerance limit. (1 G is the acceleration due to gravity.)


J. Peraire, "Variable Mass Systems: The Rocket Equation," MIT OpenCourseWare, 2004.

M. J. L. Turner, "Newton's Third Law and the Rocket Equation," Rocket and Spacecraft Propulsion, 2nd ed., New York: Springer, 2005 pp. 14–17.

M. J. L. Turner, "Launch Vehicle Dynamics," Rocket and Spacecraft Propulsion, 2nd ed., New York: Springer, 2005 pp. 115–144.

M. Voshell, "High Acceleration and the Human Body," 2004.

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.