Rocket Equation Calculator: Tsiolkovsky Delta-V, Thrust and Burn Time

How this calculator works

A rocket moves forward because it throws mass backwards at high speed. By conservation of momentum, a small mass dm thrown back at speed v_e relative to the rocket changes its velocity by dV = v_e dm / M. Adding up all the small changes as the rocket gets lighter gives the rocket equation: the velocity gained is the exhaust velocity times the natural log of the ratio of the starting mass to the final mass.

Because of the logarithm, each extra bit of velocity costs more fuel than the last. To gain a velocity equal to the exhaust velocity the rocket must burn about 63 percent of its mass, and to gain twice that, about 86 percent. This is why rockets are built in stages that drop empty tanks.

The thrust is the mass flow rate times the exhaust velocity, and the burn time is the fuel mass divided by the flow rate. The specific impulse is the exhaust velocity divided by standard gravity, in seconds. This is the ideal equation, with no gravity or air resistance.

Formulas used

• \Delta V = v_e \ln\left(\frac{M_i}{M_f}\right)
• F = \dot{m}\,v_e
• t_b = \frac{M_i - M_f}{\dot{m}}
• I_{sp} = \frac{v_e}{g_0}

Worked example

With the default numbers in the calculator above:

1. Mass ratio: 2.50
2. Velocity gained: 2748.87 m/s
3. Final velocity: 2748.87 m/s
4. Fuel used: 6000.00 kg (60.00 % of the start)
5. Specific impulse: 305.91 s
6. Thrust: 150000.00 N
7. Burn time: 120.00 s
8. Thrust to weight at the start (on Earth): 1.53
9. Starting acceleration, before gravity and drag: 15.00 m/s2

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: Velocity gain you want (0 to skip), to find the mass ratio needed 3000 m/s.

1. Mass ratio: 2.50
2. Velocity gained: 2748.87 m/s
3. Final velocity: 2748.87 m/s
4. Fuel used: 6000.00 kg (60.00 % of the start)
5. Specific impulse: 305.91 s
6. Thrust: 150000.00 N
7. Burn time: 120.00 s
8. Thrust to weight at the start (on Earth): 1.53
9. Starting acceleration, before gravity and drag: 15.00 m/s2
10. Mass ratio needed for the velocity gain you want: 2.72
11. Fuel share of the starting mass for that: 63.21 %

Common questions

Related calculators

• SUVAT Calculator: Straight-Line Motion with Constant Acceleration
• Projectile Motion Calculator

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Tsiolkovsky (1903) rocket equation; Sutton and Biblarz, Rocket Propulsion Elements; any university dynamics text (variable mass systems).

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