Newton's three laws of motion, force as a vector ($F = ma$), free-body diagrams, momentum $p = mv$ and impulse $\Delta p = F \Delta t$, and conservation of momentum in collisions

What this dot point is asking

VCAA wants you to apply Newton's three laws and the concept of momentum to motion problems, draw free-body diagrams, and use conservation of momentum in collision problems.

Newton's three laws

First law (inertia). An object at rest stays at rest, and an object in motion stays in motion at constant velocity, unless acted on by a net external force.

Second law. $F_{\text{net}} = m a$. The net force on an object equals its mass times its acceleration. Force and acceleration are vectors in the same direction.

Third law. For every action, there is an equal and opposite reaction. If A exerts a force on B, then B exerts an equal-magnitude, opposite-direction force on A.

The forces in the third law act on different bodies; they do not cancel.

Forces commonly encountered

• Gravity. $W = mg$ where $g = 9.8$ m s$^{-2}$. Always downward.
• Normal force. Perpendicular to a surface, from the surface on the object.
• Friction. Parallel to a surface, opposing relative motion. $f \leq \mu N$ for static; $f = \mu_k N$ for kinetic.
• Tension. Along a rope or string. Always pulling.
• Applied force. Pushed or pulled by something external.

Free-body diagrams

Draw the object as a dot or simple shape. Show every force acting on the object as a vector arrow originating from the dot. Label each force.

Apply Newton's second law: sum forces vectorially (component by component), set equal to $ma$.

For an object on a surface with friction:

• Normal force balances gravity perpendicular to surface.
• Net force parallel to surface = applied minus friction = $ma$.

Momentum and impulse

Momentum $\vec{p} = m \vec{v}$. Vector quantity, in the direction of velocity.

Impulse $\vec{J} = \vec{F} \Delta t = \Delta \vec{p}$. The impulse on an object equals the change in its momentum.

The impulse-momentum theorem follows from Newton's second law: $F = ma = m \Delta v / \Delta t = \Delta p / \Delta t$, so $F \Delta t = \Delta p$.

Force-time graph

The area under a force-time graph equals the impulse (= change in momentum).

For non-constant forces, integrate (or estimate area under the curve graphically).

Conservation of momentum

In an isolated system (no external forces), total momentum is conserved.

For a collision between objects 1 and 2:

where $u$ = before-collision velocity, $v$ = after-collision velocity.

Elastic vs inelastic collisions

Elastic collision. Both momentum and kinetic energy are conserved. Examples: collisions between hard balls (close to elastic in practice).

Inelastic collision. Momentum conserved; kinetic energy not (some lost to heat, sound, deformation).

Perfectly inelastic. The two objects stick together after collision. Maximum kinetic energy loss for given initial conditions.

Worked example: elastic collision

A 2 kg ball at 5 m/s collides elastically with a stationary 3 kg ball. Find final velocities.

Conservation of momentum: $2(5) + 3(0) = 2 v_1 + 3 v_2$, so $10 = 2 v_1 + 3 v_2$.

Conservation of KE: $\frac{1}{2}(2)(5)^2 = \frac{1}{2}(2) v_1^2 + \frac{1}{2}(3) v_2^2$, so $25 = v_1^2 + 1.5 v_2^2$.

From momentum: $v_2 = (10 - 2 v_1)/3$. Substitute and solve.

Result: $v_1 = -1$ m/s (ball reverses direction); $v_2 = 4$ m/s.

Check KE: $\frac{1}{2}(2)(1)^2 + \frac{1}{2}(3)(16) = 1 + 24 = 25$ J. Conserved.

Applications

Car safety. Crumple zones extend the time of a collision, reducing the force (impulse-momentum theorem: $F = \Delta p / \Delta t$). Larger $\Delta t$ means smaller $F$.

Rocket propulsion. Reaction mass expelled backwards gives forward momentum to the rocket (Newton's third law / conservation of momentum).

Sports. Follow-through in tennis or golf extends contact time to deliver more impulse.

Common errors

Confusing $F$ and $ma$. $F = ma$ relates net force to acceleration. The total force on the object equals $ma$; the acceleration is in the direction of net force.

Including internal forces. In conservation-of-momentum problems, only external forces affect total momentum. Internal forces (between parts of the system) cancel by Newton's third law.

Momentum is a vector. Direction matters. In 1D, sign indicates direction.

Confusing elastic and inelastic. Always check kinetic energy separately to determine elasticity.

Action-reaction pair on same object. The two third-law forces act on different bodies. They do not cancel; they obey Newton's third law together.

In one sentence

Newton's three laws of motion (inertia, $F = ma$, action-reaction) describe how forces produce changes in motion; the impulse-momentum theorem $J = F \Delta t = \Delta p$ relates force application over time to momentum change; momentum is conserved in any isolated system, allowing collision problems to be solved by $m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$, with elastic collisions also conserving kinetic energy and inelastic ones losing some to heat, sound or deformation.