Thrust versus weight

A rocket sitting on the pad has two main forces on it: thrust T pushing it up, and weight W = m g pulling it down (g = 9.81 m/s² on Earth). Newton’s second law says the net force makes it accelerate:

a = (T − W) / m = T/m − g

If T < W, the rocket stays on the pad, however strong the engine sounds. If T = W, the acceleration is zero: the rocket could hover, if the thrust stayed exactly equal to the weight. Only when T is bigger than W does it rise.

Engineers compare the two with the thrust-to-weight ratio:

T/W = T / (m g)

The acceleration can then be written as a = (T/W − 1) × g. A rocket with T/W = 1.5 accelerates at 0.5 g, about 4.9 m/s², at the start. One with T/W = 40 accelerates at 39 g.

The bottle rocket of the last lesson makes a good example. Its starting thrust is about 300 N. With about 0.66 litres of water (0.66 kg) and an empty mass of 0.1 kg, its mass is about 0.76 kg and its weight is 0.76 × 9.81 ≈ 7.5 N. So T/W ≈ 40 and the bottle accelerates at about 390 m/s², nearly 40 g. The water push only lasts well under a tenth of a second, so the bottle ends up moving at tens of metres per second, and then it coasts.

A real launch vehicle is the opposite. The Saturn V that sent astronauts to the Moon had a lift-off thrust of about 34 meganewtons and a lift-off mass of about 2.9 to 3.0 million kg. Its weight was about 2.9 × 10⁷ N, so T/W was only about 1.2. It rose slowly at first, with an acceleration of about 0.2 g. But the thrust went on for minutes, not a fraction of a second, and as propellant burned away the mass dropped and the acceleration climbed.

The bottle rocket is a short, hard shove. The Saturn V is a long, steady push. Both obey the same rule: a = T/m − g.