Newton's three laws of motion, inertia and momentum: WASSCE Physics

Why do passengers in a poda-poda lurch forward when the driver brakes hard? Why does a gun kick back? Why is a loaded lorry harder to stop than an empty one? Newton's three laws answer all of these, and almost every mechanics question rests on them.

Learning objectives

By the end of this lesson you should be able to:

state Newton's three laws of motion and explain inertia
define momentum and relate force to the rate of change of momentum
derive and use $F = ma$
identify action–reaction pairs correctly

Newton's first law and inertia

Definition

Newton's first law: a body continues in its state of rest or of uniform motion in a straight line unless it is acted upon by an external resultant force.

Definition

Inertia is the tendency of a body to remain at rest or, if moving, to continue moving in a straight line with uniform velocity. The mass of a body is a measure of its inertia.

Everyday examples of inertia:

Passengers jerk forward when a vehicle stops suddenly: their bodies keep moving.
Passengers jerk backward when it starts suddenly: their bodies tend to stay at rest.
Shaking a wet cloth: the cloth stops, the water keeps going.
Pulling a tablecloth quickly from under plates: the plates stay where they are.
Seat belts stop the passenger's forward motion gradually instead of the windscreen stopping it suddenly.

Momentum

Definition

The linear momentum of a body is the product of its mass and its velocity:

p = mv

It is a vector (same direction as the velocity), measured in kg m s⁻¹ (or N s).

A loaded lorry at 10 m s⁻¹ ( $m = 20\,000$ kg) has momentum $200\,000$ kg m s⁻¹; a motorcycle at the same speed ( $m = 150$ kg) has only $1500$ kg m s⁻¹. That is why the lorry is far harder to stop.

Newton's second law

Definition

Newton's second law: the rate of change of momentum of a body is directly proportional to the resultant force acting on it and takes place in the direction of the force.

For a constant mass $m$ whose velocity changes from $u$ to $v$ in time $t$ :

F \propto \dfrac{mv - mu}{t} = m\left(\dfrac{v - u}{t}\right) = ma

\begin{aligned}&F \propto \dfrac{mv - mu}{t} \\ &= m\left(\dfrac{v - u}{t}\right) \\ &= ma\end{aligned}

So $F = kma$ . The newton is defined so that $k = 1$ :

\boxed{F = ma}

Definition

One newton is the force that gives a mass of 1 kg an acceleration of 1 m s⁻².

Worked example

A car of mass 1200 kg accelerates from 10 m s⁻¹ to 25 m s⁻¹ in 6 s. Find the resultant force on it.

a = \dfrac{25 - 10}{6} = 2.5\ \text{m s}^{-2} \qquad F = ma = 1200 \times 2.5 = 3000\ \text{N}

\begin{gathered}\begin{aligned}a &= \dfrac{25 - 10}{6} \\ &= 2.5\ \text{m s}^{-2}\end{aligned} \\[6pt] \begin{aligned}F &= ma \\ &= 1200 \times 2.5 \\ &= 3000\ \text{N}\end{aligned}\end{gathered}

Worked example

A trolley of mass 4 kg is pulled along a rough floor by a horizontal force of 20 N. Friction is 8 N. Find its acceleration.

The resultant force is what accelerates it: $20 - 8 = 12$ N.

a = \dfrac{F}{m} = \dfrac{12}{4} = 3\ \text{m s}^{-2}

\begin{aligned}a &= \dfrac{F}{m} \\ &= \dfrac{12}{4} \\ &= 3\ \text{m s}^{-2}\end{aligned}

Common mistake

In $F = ma$ , $F$ is the resultant (net) force, not just the applied force. Subtract friction and other opposing forces first.

Weight

Weight is the gravitational force on a mass: applying $F = ma$ with $a = g$ ,

W = mg

A 60 kg student weighs $60 \times 10 = 600$ N on Earth. On the Moon ( $g \approx 1.6$ m s⁻²) the mass is still 60 kg but the weight is only about 96 N.

Newton's third law

Definition

Newton's third law: to every action there is an equal and opposite reaction. If body A exerts a force on body B, then B exerts an equal force on A in the opposite direction.

The two forces of an action–reaction pair:

are equal in size and opposite in direction
act on different bodies
are the same type of force (both contact forces, or both gravitational)

Action	Reaction
A swimmer pushes water backwards	water pushes the swimmer forwards
A rocket pushes exhaust gases out backwards	gases push the rocket forwards
A gun pushes a bullet forwards	the bullet pushes the gun backwards (recoil)
Your foot pushes the ground backwards	the ground pushes you forwards (walking)
The Earth pulls a mango down	the mango pulls the Earth up

Common mistake

A book resting on a table: its weight (Earth pulls book) and the table's normal reaction (table pushes book) are equal and opposite, but they are not an action–reaction pair, because both act on the same body. The reaction to the book's weight is the book pulling the Earth upwards.

Key points

First law: no resultant force → no change in velocity. Mass measures inertia.
Momentum $p = mv$ (vector, kg m s⁻¹).
Second law: force ∝ rate of change of momentum, giving $F = ma$ ; 1 N = 1 kg × 1 m s⁻².
Third law: action and reaction are equal, opposite, and act on different bodies.

WASSCE examination focus

WAEC tests this

Expect to state all three laws precisely (word-perfect statements earn full marks), to explain an everyday observation using inertia, to show how the second law leads to $F = ma$ , and to calculate with $F = ma$ using the resultant force. Objective items often ask which pair of forces is an action–reaction pair.

Quick check

Which property of a body measures its inertia?
A 50 kg box is pushed with a 300 N force against 100 N of friction. Find its acceleration.
Why does a gun recoil when fired?

Answers: 1. Its mass. 2. $(300 - 100) \div 50 = 4$ m s⁻². 3. By Newton's third law, the gun pushes the bullet forward, so the bullet pushes the gun backward with an equal force.

Newton's three laws of motion, inertia and momentum

Learning objectives

Newton's first law and inertia

Momentum

Newton's second law

Weight

Newton's third law

Key points

WASSCE examination focus

Quick check

Keep going with newton's laws of motion

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