Force and Laws of Motion • Topic 3 of 3

Newton's Third Law & Conservation of Momentum

Newton's Third Law. To every action there is an equal and opposite reaction. Whenever one body exerts a force (the action) on a second body, the second body simultaneously exerts an equal force in the opposite direction (the reaction) on the first. The two forces are equal in magnitude and opposite in direction, but they always act on two different bodies. Because they act on different bodies, action and reaction forces never cancel each other.

Everyday examples:

When you walk, your foot pushes the ground backward (action) and the ground pushes you forward (reaction).
A swimmer pushes water backward and the water pushes the swimmer forward.
A rocket pushes hot gases downward and the gases push the rocket upward.
A gun recoils backward when it fires a bullet forward.

Conservation of momentum. When two or more bodies interact and no external unbalanced force acts on the system, the total momentum of the system stays constant. This is the law of conservation of momentum: the total momentum before an interaction equals the total momentum after it. It follows directly from Newton's Third Law.

For two bodies $A$ and $B$ with masses $m_1$ and $m_2$, initial velocities $u_1$ and $u_2$, and final velocities $v_1$ and $v_2$:

$m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$ (total momentum is conserved).

Recoil of a gun. Before firing, both the gun and bullet are at rest, so the total momentum is zero. After firing, the bullet moves forward with momentum $m_b v_b$. To keep the total momentum zero, the gun must move backward with an equal and opposite momentum $m_g v_g$. Since the gun's mass $m_g$ is much larger than the bullet's mass $m_b$, the recoil velocity $v_g$ of the gun is small.

Collisions. In a collision, momentum is always conserved (provided no external force acts). If two bodies stick together after impact (a perfectly inelastic collision), they move with a single common velocity that can be found from the conservation equation. This is how we analyse two trolleys joining together, a ball of clay hitting another, or wagons coupling in a railway yard.

Solved Examples

Worked Example

When a swimmer pushes water backward, she moves forward. Identify the action and reaction forces and state which law applies.

Solution

Action: the swimmer's hands push the water backward.
Reaction: the water pushes the swimmer forward with an equal and opposite force.
These act on two different bodies (water and swimmer).

Answer: This is Newton's Third Law; the forward reaction of the water moves the swimmer.

Worked Example

A gun of mass $4\ \text{kg}$ fires a bullet of mass $0.02\ \text{kg}$ with a velocity of $200\ \text{m/s}$. Find the recoil velocity of the gun.

Solution

Total momentum before firing $= 0$ (both at rest).
By conservation: $m_g v_g + m_b v_b = 0$, so $m_g v_g = -m_b v_b$.
$v_g = \frac{-m_b v_b}{m_g} = \frac{-0.02 \times 200}{4} = -1\ \text{m/s}$.

Answer: The gun recoils at $1\ \text{m/s}$ in the direction opposite to the bullet.

Worked Example

A $2\ \text{kg}$ trolley moving at $3\ \text{m/s}$ collides with and sticks to a stationary $1\ \text{kg}$ trolley. Find their common velocity after the collision.

Solution

Total momentum before $= (2 \times 3) + (1 \times 0) = 6\ \text{kg m/s}$.
After collision they move together with mass $(2 + 1) = 3\ \text{kg}$ and common velocity $v$.
By conservation: $6 = 3v \Rightarrow v = 2\ \text{m/s}$.

Answer: The combined trolleys move at $2\ \text{m/s}$.

Worked Example

Why does action and reaction not cancel each other even though they are equal and opposite?

Solution

Action and reaction are equal in magnitude and opposite in direction.
However, they act on two different bodies, not on the same body.
Forces cancel only when they act on the same body.

Answer: Because they act on two different bodies, action and reaction never cancel each other.

Worked Example

Two objects of masses $100\ \text{g}$ and $200\ \text{g}$ move toward each other at $2\ \text{m/s}$ and $1\ \text{m/s}$ respectively along the same line. After collision they stick together. Find their common velocity.

Solution

Convert: $m_1 = 0.1\ \text{kg}$, $u_1 = +2\ \text{m/s}$; $m_2 = 0.2\ \text{kg}$, $u_2 = -1\ \text{m/s}$ (opposite direction).
Total momentum before $= (0.1 \times 2) + (0.2 \times -1) = 0.2 - 0.2 = 0\ \text{kg m/s}$.
After: $(0.1 + 0.2)v = 0 \Rightarrow v = 0\ \text{m/s}$.

Answer: The stuck-together masses come to rest, $v = 0\ \text{m/s}$.

Worked Example

A rocket works by burning fuel and ejecting hot gases. Explain how it gains forward motion using Newton's Third Law and conservation of momentum.

Solution

The rocket pushes hot gases downward at high speed (action).
By Newton's Third Law, the gases push the rocket upward with an equal force (reaction).
The downward momentum of the gases equals the upward momentum gained by the rocket, conserving total momentum.

Answer: The reaction of the ejected gases drives the rocket forward, and total momentum stays conserved.

Key Points

Newton's Third Law: every action has an equal and opposite reaction, acting on two different bodies.
Action and reaction never cancel because they act on different bodies.
Law of conservation of momentum: with no external force, total momentum stays constant.
For two bodies: $m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$.
Recoil of a gun, rockets and swimming are explained by the Third Law and momentum conservation.

Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.Newton's Third Law states that action and reaction are:

Explanation: They are equal and opposite but act on two different bodies.

Q2.Action and reaction forces do not cancel each other because they:

Explanation: Forces cancel only when they act on the same body.

Q3.In the absence of an external force, the total momentum of a system is:

Explanation: This is the law of conservation of momentum.

Q4.A gun recoils when fired because of conservation of:

Explanation: The forward momentum of the bullet equals the backward momentum of the gun.

Practice more MCQs → 📝 Assignment · 30 marks