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Thrust-to-weight ratio

Rockets and landing · level 5, lesson 2 of 6

A question to start

A rocket built to leave Mars has a thrust-to-weight ratio of 1.5 when it sits on its pad on Mars. Could it lift off from Earth?

The idea

The thrust-to-weight ratio (TWR) is

\[\text{TWR} = \frac{T}{mg}.\]

It tells you at a glance what the engines can do:

  • TWR \(> 1\): the rocket can lift off or climb.
  • TWR \(= 1\): it can hover.
  • TWR \(< 1\): it cannot leave the ground (though it can still speed up in space, where weight does not hold it back).

Starting acceleration. For vertical flight, \(ma = T - mg\), so

\[a = g\,(\text{TWR} - 1).\]

A rocket with TWR 1.3 on Earth starts upwards at \(9.81 \times 0.3 = 2.9\) m/s².

On another planet. TWR depends on \(g\). The same rocket has

\[\text{TWR}_\text{Mars} = \text{TWR}_\text{Earth} \times \frac{g_E}{g_M} = 2.64 \times \text{TWR}_\text{Earth}.\]

So the Mars rocket in the question, at 1.5 on Mars, has \(1.5/2.64 = 0.57\) on Earth: it could not lift off from home.

A rocket with \(T/m\) = 15 m/s² Mars Earth Moon
TWR 4.0 1.5 9.3
Starting acceleration upwards 11.3 m/s² 5.2 m/s² 13.4 m/s²

Why not a huge TWR? Bigger engines are heavier, and a very high TWR crushes the crew and payload with acceleration. Launchers on Earth usually start around 1.2 to 1.5; upper stages, already in space, often have TWR below 1. Landers need TWR a bit above 1 so they can slow down, and must be able to throttle down to hover.

Worked examples

1. Saturn V. \(T = 34\) MN, \(m = 2.97 \times 10^6\) kg: TWR \(= 34\times10^6 / (2.97\times10^6 \times 9.81) = 1.17\); \(a = 9.81 \times 0.17 = 1.7\) m/s².

2. The Apollo lunar module's ascent stage. Thrust about 16 kN, mass about 4,700 kg, on the Moon: TWR \(= 16{,}000/(4{,}700 \times 1.62) = 2.1\). The same stage on Mars: \(16{,}000/(4{,}700 \times 3.71) = 0.92\): it could not have left Mars.

3. A throttling lander. A 1,000 kg lander on Mars must hover. It needs \(T = 3{,}710\) N. If its engines can give between 2,000 and 12,000 N, it can hover, slow down hard, or fall a little faster.

Try it

  1. A rocket on Earth has TWR 1.4. Find its starting acceleration.
  2. The same rocket on Mars: find its TWR and starting acceleration.
  3. A lander of 3,000 kg must be able to decelerate at 2 m/s² while descending on Mars. What thrust does it need?
  4. Why do upper stages not need TWR above 1?
  5. A Mars ascent rocket must lift 20 t from the surface with TWR 1.3 on Mars. What thrust does it need?
Answers
  1. \(9.81 \times 0.4 = 3.9\) m/s².
  2. TWR \(= 1.4 \times 2.64 = 3.7\); \(a = 3.71 \times 2.7 = 10.0\) m/s².
  3. \(T = m(g + a) = 3000 \times 5.71 = 17{,}100\) N.
  4. They fire already in space, moving sideways near orbital speed; they do not have to hold themselves up against gravity as at lift-off.
  5. \(T = 1.3 \times 20{,}000 \times 3.71 = 96{,}500\) N, about 97 kN.

Common mistake

Thinking TWR is a property of the engine alone. It depends on the whole mass and on the planet's \(g\). As propellant burns, \(m\) falls and TWR rises; on another planet it changes by the ratio of the \(g\) values.

Where it leads