Chapter 1 · Mechanics
1-8 The Law of Conservation of Momentum
❓ Lesson question
When you step from a stationary small boat onto a dock, why does the boat move backward, and what determines the boat’s speed?
You will learn to
- 1Explain why the total momentum of an isolated system remains constant.
- 2Apply conservation of momentum to calculate unknown velocities before and after a collision.
- 3Define the system, choose an axis and state the positive direction when applying conservation of momentum.
You finish a trip on a small boat and step forward onto the stone dock. As your foot moves forward, the boat slides backward, increasing the gap between the boat and the dock. Step gently and it moves away slowly, whereas a stronger push gives it a greater backward speed. No one else pushes the boat — yet it still moves backward.
✍️ Predict before you continue
Write your answer before reading on. A prediction you got wrong and then examined teaches you more than an answer you were handed.
(a) During the push-off, do you and the boat undergo equal or different changes in momentum magnitude?
(b) An empty boat and a boat loaded with crates receive the same impulse. Which boat recoils with the greater speed?
🔬 Explore — Recoil when two students push apart
What you’ll do: Measure how far two students on wheeled chairs move after pushing apart, and investigate how the distance each moves depends on their mass.
- 1Two students sit facing each other on the chairs, palms pressed together, both at rest on the start strip.
- 2On a count, both students push firmly once against each other’s palms, then keep their hands still as the chairs roll apart. Mark where each chair stops.
- 3Repeat with a heavier student facing a lighter one, from the same start strip.
- 4Repeat once more, but this time only one student pushes while the other keeps their arms rigid. Mark the stopping points again.
Analyse what you collected. (a) When the two students have different masses, which one moves further, and how does the distance depend on their mass? (b) When only one student pushed, did only that chair move, or did both? What does that tell you about the forces during an interaction?
In Explore, the chair carrying the lighter student travelled farther, so it left the push with the greater speed — and, notably, both chairs moved apart even when only one student pushed. One might expect that only the chair that is pushed should move, yet the observations show that both chairs move apart.
The two chairs do not act independently; they interact as a pair. Whenever one student pushes on the other, the other pushes back with a force equal in magnitude and opposite in direction, acting for exactly the same time. Each chair therefore receives an impulse equal in magnitude but opposite in direction, so each undergoes an equal and opposite change in momentum; because it has the smaller mass, the lighter chair moves away at the greater speed.
🎛️ Simulation — conservation of momentum
Speeds — different, and that is what you see
A: 1.50 m/s forward
B: 0.75 m/s backward
The lighter body always ends up faster. With p = mv fixed, small m forces large v.
Momentum — equal and opposite, whatever you do
total = +0.00 kg·m/s
Drag any slider. The two bars are always the same length. The total is always zero, because it started at zero and no outside force acted.
This answers the prediction on p.44 (a): the changes in momentum are equal in magnitude, not different — even though the speeds are obviously different. And it answers the Explore question on p.45: there is no slider here for "who pushed", because a push is never one-sided. Whichever student pushes, both chairs receive the same size of impulse in opposite directions, so both move.
A group of objects under consideration is called a system. The forces exerted on each other within the system are called internal forces, and the forces exerted by agents external to the system are called external forces.
The total momentum of the isolated system is the same before and after the interaction. This principle is called the law of conservation of momentum. When object A with mass m1 and velocity v1 collides with object B with mass m2 and velocity v2, and their respective velocities become v1′ and v2′:
total momentum before = total momentum after
The law holds when the system receives no net external impulse. It holds too if external forces such as gravity are received but can be ignored. For motion in a plane, resolve the vectors into x and y directions and apply the law independently in each.
Point
- • A group of objects under consideration is a system. Forces between members are internal; forces from outside are external.
- • The total momentum of an isolated system does not change during a collision:m1v1+m2v2=m1v1′+m2v2′.
- • It holds when no impulse is received from external forces. In a plane, resolve intox and y and conserve each independently.
📐 Equation box
Conservation along an axis (two bodies)
Momentum of a body
🛠️ Worked example (book p.47)
Situation
(a) Object A of mass 1.2 kg moving in the positive direction at 3.0 m/s and object B of mass 2.8 kg moving in the negative direction at 2.0 m/s along a straight line collide and stick together. Determine the velocity after they stick together.
(b) On a smooth horizontal surface, object A of mass 0.20 kg moving at 2.0 m/s collides with object B of mass 0.80 kg that is at rest. After the collision A moves perpendicular to its original direction and B moves off at 30° to it. Determine the final speeds of A and B.
In a New Context
Rocket propulsion. A rocket operates by expelling exhaust gases backward at high velocity. The gases gain backward momentum, while the rocket gains equal forward momentum. For the rocket-and-exhaust system, if external impulse is negligible, total momentum is conserved: Δprocket=−Δpexhaust. During launch, gravity acts downward, so the rocket accelerates upward only when its thrust is greater than its weight.
Explain why a rocket can accelerate in the vacuum of space even though there is no external medium to push against.
Answer. Because a rocket does not push against anything outside itself. The system is the rocket plus its fuel, and it begins with a certain total momentum. When the engine throws mass backward at high speed, that exhaust carries backward momentum, so the remaining rocket must carry an equal forward momentum for the total to stay unchanged. The air is not a party to the interaction — in fact air only gets in the way, so a rocket works better in vacuum. The common phrase "pushing against the air" describes an aeroplane propeller, not a rocket.
🧪 Nature of Science — Evidence — testing a hypothesis against conservation laws
In 1930, measurements of beta decay appeared to show that some energy and momentum were missing from the observed products. Rather than immediately abandoning the well-tested conservation laws, Wolfgang Pauli proposed that an undetected neutral particle carried away the missing energy and momentum. This particle later became known as the neutrino. Pauli's proposal made a testable prediction: if the particle existed, experiments might eventually detect evidence of its interactions. In 1956, Frederick Reines and Clyde Cowan reported the experimental detection of the neutrino.
This episode shows that established laws are not accepted without question. Scientists retain them while the evidence strongly supports them, but they also develop and test alternative explanations when new observations appear inconsistent with those laws.
An answer to "your turn". Because a hypothesis that predicts something can be checked, and a declaration of failure predicts nothing. Saying "conservation is wrong here" closes the question and gives no experiment to perform; saying "an unseen neutral particle carries the balance" says what to look for, what mass and charge it must have, and what would count as finding it. It also risks something: had the missing momentum not matched a single consistent particle, the idea would have died. Twenty-six years later the prediction was confirmed — which it could not have been if Pauli had merely announced a failure.
Your turn: Why was proposing a testable hypothesis about a particle more scientifically useful than simply declaring that the conservation laws had failed?
✅ Lesson question, answered
When you step from a stationary small boat onto a dock, why does the boat move backward, and what determines the boat’s speed?
Treat the person and the boat as one system. Initially the total momentum is zero. During the short push-off, the external horizontal impulse is negligible — the water exerts very little horizontal force in that fraction of a second — so momentum is conserved:
The person and the boat therefore gain equal and opposite momentum. Sincep=mv, the boat's speed is fixed by two things and only two: the size of the push (which sets the shared momentum magnitude) and the boat's mass. A lighter boat moves faster than a heavier one for the same momentum, which is why an empty boat shoots away and a loaded one barely drifts.
- system
- — The group of objects being considered together.
- internal force
- — A force exerted by one member of the system on another.
- external force
- — A force exerted on the system by something outside it.
- isolated system
- — A system receiving zero net external impulse, so its total momentum is constant.
How the real paper is built
The ministry sets 50% multiple choice and 50% written answer. They test different things: the MCQ half rewards eliminating fast and spotting a distractor built from a plausible wrong method; the written half asks you to show that you understand why. Practise both halves — being fluent at one and weak at the other costs you exactly half the paper.
One step. You are given the formula; put the numbers in.
Two steps, or you must choose the right component before you can start.
Multi-step, or a case the textbook does not set and examiners do.
📝 Exam-style question
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