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F = ma

Rockets and landing · level 5, lesson 1 of 6

A question to start

A lander of mass 1,000 kg is descending on Mars. Its engines give 5,000 N of thrust upwards. Is it speeding up or slowing down, and by how much each second?

The idea

Newton's three laws of motion (1687):

  1. First law. A body stays at rest, or moves at a steady speed in a straight line, unless a net force acts on it. (Things do not slow down "naturally"; friction and drag slow them.)
  2. Second law. The net force on a body equals its mass times its acceleration: $\(F_\text{net} = m a.\)$ Force in newtons, mass in kg, acceleration in m/s². One newton is the force that gives 1 kg an acceleration of 1 m/s².
  3. Third law. When A pushes on B, B pushes on A with an equal force in the opposite direction.

Free-body diagrams. Draw the body as a dot, and every force on it as an arrow. Add them (taking one direction as positive) to get \(F_\text{net}\), then \(a = F_\text{net}/m\).

thrust 5,000 Nweight mg = 3,710 N

The lander. Weight on Mars: \(mg = 1000 \times 3.71 = 3{,}710\) N down. Thrust 5,000 N up. Taking up as positive: \(F_\text{net} = 5000 - 3710 = 1{,}290\) N up, so \(a = 1290/1000 = 1.29\) m/s² upwards. The lander is moving down, so an upward acceleration means it is slowing down, by 1.29 m/s every second.

Weight is a force like any other, \(W = mg\). Mass is the same everywhere; \(g\) changes.

Worked examples

1. The lander on Earth. The same 1,000 kg lander, the same 5,000 N, but on Earth: weight 9,810 N. \(F_\text{net} = 5000 - 9810 = -4{,}810\) N, \(a = -4.81\) m/s²: it speeds up downwards. Those engines cannot land it on Earth.

2. Pushing a crate in orbit. An astronaut pushes a 200 kg crate with 50 N. \(a = 50/200 = 0.25\) m/s². After 4 s the crate moves at 1 m/s, and it keeps going (first law) until something stops it. Stopping it takes the same push the other way: mass resists changes of motion even where nothing weighs anything.

3. A rover on a slope. A 1,025 kg rover's wheels push it forward with 800 N on flat ground; the soil's drag on the wheels is 600 N. \(a = 200/1025 = 0.2\) m/s².

Try it

  1. A 2,000 kg lander on Mars has 10,000 N of thrust. Find its acceleration and say whether a descending lander is slowing.
  2. What thrust lets that lander hover on Mars?
  3. A 50 kg box sits on the floor of a lift accelerating upwards at 2 m/s² on Earth. How hard does the floor push on it?
  4. Use the third law: what does a rocket push on, and what pushes the rocket?
  5. A ball floats in the space station. Why does it keep drifting after you let go?
Answers
  1. Weight \(= 7{,}420\) N. \(F_\text{net} = 2{,}580\) N up, \(a = 1.29\) m/s² up. Yes, slowing.
  2. \(2000 \times 3.71 = 7{,}420\) N.
  3. \(N - mg = ma\), so \(N = 50 \times (9.81 + 2) = 590\) N.
  4. It pushes its exhaust backwards; the exhaust pushes it forwards.
  5. No net force acts on it (first law), so it keeps the velocity it had.

Common mistake

Using thrust as the net force. Weight, drag and other forces act at the same time. \(F = ma\) needs the net (total) force.

Where it leads