Details

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

symbols:

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.)

References:

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.