Forces and momentum

Newton's laws connect the forces on an object to its motion.

First law (inertia). An object continues at rest, or moving with constant velocity (constant speed in a straight line), unless acted on by a resultant force. So constant velocity ⇔ balanced forces (resultant = 0). This is why terminal velocity is constant: drag has grown to balance weight.

Second law. The resultant force on an object equals its rate of change of momentum: $F_{\text{net}} = \frac{\Delta p}{\Delta t}$ For constant mass this simplifies to the familiar $F_{\text{net}} = ma$ The acceleration is in the same direction as the resultant force and is proportional to it.

Third law. If body A exerts a force on body B, then body B exerts an equal-sized, opposite-direction force on body A. The two forces:

• are the same type (both gravitational, both contact, etc.),
• act on two different bodies, and
• therefore never cancel (they act on different objects, so you never add them on one free-body diagram).

A worked distinction that earns marks: a book resting on a table. Its weight (Earth pulls book down) and the normal force (table pushes book up) are balanced — but they are not a Newton's-third-law pair, because they act on the same body (the book) and are different types. The third-law partner of the book's weight is the book pulling the Earth up; the partner of the normal force is the book pushing the table down.

A free-body diagram isolates one object and shows all the forces acting on it as labelled arrows, drawn from the object and pointing in the direction of each force. Longer arrow = bigger force.

The forces you must recognise:

Force	Symbol	Description
Weight	$W = mg$	Gravitational pull, always vertically down
Normal (reaction)	$N$ or $R$	Contact force perpendicular to a surface
Friction	$f$	Contact force parallel to a surface, opposing (attempted) motion
Tension	$T$	Pull along a rope, string or cable
Drag / air resistance	$F_d$	Resistive force opposing motion through a fluid, grows with speed
Buoyancy (upthrust)	$F_b$	Upward force from a fluid on a submerged/floating object
Applied / thrust	$F$	Any push or pull applied directly

Friction detail (SL): static friction adjusts to prevent motion up to a maximum $f_{\max} = \mu_s N$; once the object slides, kinetic (dynamic) friction is $f = \mu_d N$, where $\mu$ is the coefficient of friction. Friction always opposes the relative sliding (or the tendency to slide).

Method for any dynamics problem:

1. Draw the free-body diagram for the object.
2. Choose axes (often along and perpendicular to the motion / slope).
3. Resolve forces onto the axes.
4. Apply $F_{\text{net}} = ma$ along each axis (with $a = 0$ if in equilibrium).

Linear momentum is defined as $p = mv$ It is a vector (same direction as the velocity) with units kg m s⁻¹ (equivalently N s).

Newton's second law is really a statement about momentum: $F_{\text{net}} = \frac{\Delta p}{\Delta t}$

Impulse is the change in momentum an object experiences: $J = F\,\Delta t = \Delta p = mv - mu$

• Impulse has units N s (= kg m s⁻¹).
• On a force–time graph, the impulse = area under the graph. For a varying force, use the area (not the peak force).

Why impulse matters in real life (a favourite exam context): to reduce the force in an impact, increase the contact time. The change in momentum $\Delta p$ is fixed (the object still has to stop), so a longer $\Delta t$ means a smaller average force $F = \Delta p / \Delta t$. This is the physics of:

• crumple zones and airbags in cars,
• bending your knees when you land,
• a cricketer moving their hands back to catch a ball.

Two short derivations turn these laws from things-to-memorise into things-you-understand — exactly the reasoning IB rewards at the top grades.

1. F = ma is a special case of F = Δp/Δt. Start from the true form of Newton's second law: $F = \frac{\Delta p}{\Delta t} = \frac{\Delta(mv)}{\Delta t}$ If the mass is constant, it comes outside the change: $F = m\,\frac{\Delta v}{\Delta t} = ma \quad(\text{since } a = \Delta v/\Delta t)$ So $F = ma$ is just the constant-mass version. The momentum form is more general — it still works for systems where mass changes (e.g. a rocket burning fuel), which is why the data booklet form is written with momentum.

2. Conservation of momentum comes from Newton's third law. Consider two bodies, A and B, that interact (collide) for a time $\Delta t$. By Newton's third law the forces are equal and opposite at every instant: $F_{A\to B} = -\,F_{B\to A}$ They act for the same time $\Delta t$, so the impulses are equal and opposite: $F_{A\to B}\,\Delta t = -\,F_{B\to A}\,\Delta t \quad\Rightarrow\quad \Delta p_B = -\,\Delta p_A$ Therefore the momentum gained by B equals the momentum lost by A, so the total momentum change of the system is zero: $\Delta p_A + \Delta p_B = 0 \quad\Rightarrow\quad p_{\text{total}}\ \text{is constant.}$

This is why momentum is conserved in every collision and explosion (with no external force): it is a direct consequence of the third law. Knowing this lets you answer "explain why momentum is conserved" questions — a frequent 2–3 mark item — with confidence rather than just asserting it.

Micro-example of the impulse idea: a 0.50 kg ball dropped onto the floor arrives at $5.0\,\text{m s}^{-1}$ and rebounds at $4.0\,\text{m s}^{-1}$. Taking up as positive, $\Delta p = m v - m u = 0.50(+4.0) - 0.50(-5.0) = 2.0 + 2.5 = 4.5\,\text{N s}$ upward. The floor delivered an upward impulse of 4.5 N s — larger than either momentum alone, because the ball's direction reversed.

Principle of conservation of linear momentum: in a closed system (no external resultant force), the total momentum before an interaction equals the total momentum after. It follows directly from Newton's third law and is one of the most powerful tools in mechanics.

For two bodies in 1D: $m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$ Always assign signs for direction (e.g. right = +, left = −).

Types of collision:

Type	Momentum	Kinetic energy	Example
Elastic	conserved	conserved	ideal gas molecules; snooker balls (nearly)
Inelastic	conserved	not conserved (some → heat/sound/deformation)	most real collisions
Perfectly inelastic	conserved	maximum KE lost; objects stick together	clay ball hitting a block

Explosions are the reverse of a perfectly inelastic collision: an object initially at rest has zero total momentum, so after the explosion the fragments must have momenta that add (as vectors) to zero — e.g. a gun and bullet recoil in opposite directions with equal-magnitude momentum.

Checking whether a collision is elastic: compute the total kinetic energy ($\tfrac{1}{2}mv^2$ for each body) before and after. If it is unchanged, the collision is elastic; if it has decreased, it is inelastic. Momentum is conserved either way.

An object moving in a circle at constant speed is still accelerating, because its velocity (a vector) is continually changing direction. This acceleration — and the resultant force causing it — points towards the centre of the circle and is called centripetal.

Key relationships: $a = \frac{v^2}{r} = \omega^2 r, \qquad F_{\text{net}} = \frac{mv^2}{r} = m\omega^2 r$ where $\omega = \dfrac{2\pi}{T} = 2\pi f$ is the angular velocity (rad s⁻¹) and $v = \omega r$.

What provides the centripetal force? It is not a new force — it is the name for the resultant of the real forces, always directed to the centre:

• a car turning a corner → friction between tyres and road,
• a planet orbiting the Sun → gravity,
• a ball on a string → tension,
• a bucket swung in a vertical circle → tension + a component of weight.

Two facts examiners love:

1. The centripetal force does no work and the speed stays constant, because the force is always perpendicular to the velocity.
2. The so-called "centrifugal force" (feeling thrown outward) is a fictitious force experienced only in the rotating frame — it must never appear on a free-body diagram drawn in an inertial frame.