Third Law and Conservation of Momentum: MDCAT Physics notes

Newton's third law

To every action there is always an equal and opposite reaction. If body A exerts a force on body B, then B exerts an equal and opposite force on A.

• Action and reaction are equal in magnitude and opposite in direction.
• They have the same line of action and are of the same nature (both contact, both gravitational, etc.).
• They act on different bodies, so they never cancel each other. The third law always holds.
• Two bodies in contact push each other in opposite directions.

Everyday examples

• Walking / horse pulling a cart: the horse pushes backward on the ground; the ground's forward reaction on the horse moves it (and the cart) forward.
• Rowing a boat: the oars push water backward; water pushes the boat forward.
• Rocket and jet: exhaust gases are pushed backward; the gases push the rocket forward.
• Gun recoil: the gun pushes the shell forward, the shell pushes the gun backward.

Law of conservation of momentum

The total linear momentum of an isolated system remains constant. An isolated system is one on which no net external force acts. Internal forces (action-reaction pairs inside the system) cancel in pairs, so they cannot change the total momentum.

$$m_1\vec u_1 + m_2\vec u_2 = m_1\vec v_1 + m_2\vec v_2$$

Conservation of momentum is a consequence of Newton's third law.

Recoil of a gun

Before firing, total momentum is zero. After firing, $m_b v_b + M v_g = 0$, so

$$v_g = -\frac{m_b v_b}{M}$$

The negative sign shows the gun moves backward. A heavy gun recoils slowly.

Bodies that stick together

A bullet of mass $m$ at speed $u$ embeds in a block $M$ at rest: $v = mu/(m+M)$. Example: $0.02\ \text{kg}$ at $300\ \text{m s}^{-1}$ into a 2 kg block gives $v = 6/2.02 \approx 3\ \text{m s}^{-1}$.

A bird swooping on prey: the bird and prey together form the system; momentum of the pair is conserved, not of either alone.

Zero total momentum

Momentum is a vector. A group of identical birds can have zero total momentum only if they fly in different directions so their momenta cancel. Birds flying the same way can never have zero total momentum.

Key formulas

• $\vec F_{AB} = -\vec F_{BA}$
• $\sum \vec p_{before} = \sum \vec p_{after}$ (isolated system)
• Recoil: $v_g = -m_b v_b/M$
• Sticking: $v = (m_1u_1 + m_2u_2)/(m_1+m_2)$

Common MDCAT traps

• "Action and reaction cancel each other" is false; they act on different bodies.
• Momentum is conserved in an isolated system, not merely any closed or open system.
• A horse moves because of the ground's force on the horse, not the horse's force on the cart.
• Recoil is explained by Newton's third law (and momentum conservation), not energy conservation.

Quick revision

• Action = reaction, on different bodies.
• Same nature, same line of action, opposite direction.
• Internal forces cannot change total momentum.
• Gun and bullet: equal and opposite momenta.
• Boat moves forward by pushing water backward.