NDA Physics · Laws of Motion and Forces

Newton's Three Laws of Motion

Newton's three laws: a body keeps its state of motion unless a net force acts (inertia), force equals rate of change of momentum (F = ma), and every action has an equal and opposite reaction.

All Laws of Motion and Forces notes NDA Physics strategy

Why this matters

This is the heart of the chapter and its largest subtopic — roughly 18 PYQs across 2018–2026. Most are one-line recall (inertia, mass vs weight, what stays constant at uniform velocity) or a single F = ma substitution; the only HARD pocket is combining two forces into a resultant via the parallelogram law. Drill F = ma, the parallelogram formula, and the mass-vs-weight distinction and you clear almost the whole subtopic.

Concept 1 of 6

First law — inertia

Intuition

Left alone, things keep doing what they are doing: a body at rest stays at rest, and a body in motion keeps moving in a straight line at constant speed, UNLESS a net external force acts. This reluctance to change motion is inertia, and it is measured by mass — the heavier the body, the more inertia it has and the harder it is to start, stop, or turn.

Definition

Newton's first law (law of inertia): a body continues in its state of rest or of uniform motion in a straight line unless acted on by a net external force. Inertia is the tendency of a body to resist any change in its state of motion; it is measured by mass. More mass = more inertia. A direct consequence: at uniform velocity the acceleration is zero, so the net force is zero.

Condition for the first law (equilibrium of motion)

\vec{F}_{\text{net}} = 0 \iff \vec{a} = 0

F_netnet (resultant) external force on the body
aacceleration; zero means rest or constant velocity

Worked example

A car cruises down a straight road at a steady 60 km/h. What is its acceleration, and what is the net force on it?

Steady speed in a straight line means the velocity is constant.
Constant velocity means the acceleration is zero.
By the first law (and F = ma), zero acceleration means the net force is zero — the engine's driving force exactly balances drag and friction.

Answer:Acceleration = 0; net force = 0.

Practice this conceptself-check · 4 quick reps

Try it yourself

A cricket ball and a tennis ball are thrown at the same speed. Which is harder to stop, and why?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
What physical quantity measures inertia?
2.
At uniform velocity, what is a body's acceleration?
3.
Of a cricket ball and a feather, which has more inertia?
4.
A body moves at constant velocity. What is the net force on it?

From the bank · past-year question

Example 1Laws of Motion and ForcesEASY

At uniform speed the acceleration is

[Q150 · Apr · 2025]

Constant velocity does NOT mean changing speed

NDA 2018 asked which statement is NOT correct for a body moving at constant velocity. The wrong statement is "its speed changes with time" — at constant velocity the speed is fixed, acceleration is zero, and the net force is zero. Don't confuse constant velocity (vector) with merely constant speed; here both are fixed.

Drill 5 more on first law — inertia

Concept 2 of 6

Second law — F = ma

Intuition

Push a body and it accelerates: the harder you push, the faster it speeds up, and the heavier it is, the less it speeds up for the same push. The precise statement is that force equals the rate of change of momentum, which for constant mass reduces to the familiar F = ma. Mass is the constant of proportionality between force and acceleration.

Definition

Newton's second law: the net force on a body equals the rate of change of its momentum, $\vec{F} = \dfrac{d\vec{p}}{dt}$ . For constant mass this reduces to $\vec{F} = m\vec{a}$ . Here mass is the constant of proportionality between the applied force and the resulting acceleration; force and acceleration always point in the same direction. To apply it, draw a free-body diagram (every force ON the body as an arrow), take the vector sum, and set it equal to $m\vec{a}$ .

Newton's second law

\vec{F} = \frac{d\vec{p}}{dt} = m\vec{a} \quad (\text{constant } m)

Fnet force (N)
p = mvlinear momentum (kg m/s)
mmass (kg) — the constant of proportionality
aacceleration (m/s²)

Draw every force acting ON the block as an arrow from its centre. The vertical pair (N up, mg down) cancels on flat ground; the net horizontal force F minus f gives the acceleration via F-net = ma.

Worked example

A 1500 kg car travelling at 20 m/s is brought to rest by braking in 4 s. What is the magnitude of the braking force?

Find the acceleration: $a = \dfrac{v - u}{t} = \dfrac{0 - 20}{4} = -5\,\text{m/s}^2$ .
The magnitude of the deceleration is $5\,\text{m/s}^2$ .
Apply $F = ma = 1500 \times 5 = 7500\,\text{N}$ .

Answer:7500 N (7.5 kN).

Practice this conceptself-check · 4 quick reps

Try it yourself

A net force of 5 N acts on a 10 kg mass. What acceleration does it produce?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
State Newton's second law in its momentum form.
2.
What acceleration does a 4 N force give a 2 kg body?
3.
In F = ma, which quantity is the constant of proportionality?
4.
A 0.5 m/s² acceleration on a 10 kg mass needs what force?

From the bank · past-year question

Example 2Laws of Motion and ForcesMODERATE

A car weighs 1000 kg moving at 72 km/h. Driver brakes; car stops in 0.2 s. The retarding force is

[Q96 · Sep · 2024]

Force is proportional to rate of change of momentum, NOT to momentum itself

NDA 2025 asked which second-law statement is NOT correct. The wrong one says "net force is proportional to the body's momentum" — it should be proportional to the RATE OF CHANGE of momentum (dp/dt). A body can have huge momentum yet zero net force (constant velocity).

Watch the units when computing F = ma

Convert km/h to m/s before using F = ma (divide by 3.6: 72 km/h = 20 m/s). NDA 2024 gave 72 km/h and a 0.2 s stop, yielding a = 100 m/s² and F = 100 kN — the large answer is a sign you converted correctly, not an error.

Drill 3 more on second law — f = ma

Concept 3 of 6

Third law — action and reaction

Intuition

Forces always come in pairs. When you push on a wall, the wall pushes back on you with an equal force in the opposite direction. A rocket pushes gas backward and the gas pushes the rocket forward. The two forces are equal in size, opposite in direction, and crucially act on DIFFERENT bodies.

Definition

Newton's third law: to every action there is an equal and opposite reaction. The action and reaction forces are equal in magnitude, opposite in direction, and act on two different bodies. Because they act on different bodies, they never cancel each other — cancellation would require both forces on the SAME body.

Newton's third law (force pair)

\vec{F}_{AB} = -\vec{F}_{BA}

F_ABforce exerted by A on B (action)
F_BAforce exerted by B on A (reaction)

The action and reaction are equal and opposite, but they act on two different objects (one on B, one on A). That is why they do not cancel each other — cancellation needs both forces on the SAME body.

Worked example

A swimmer pushes the water backward with a force of 200 N. With what force, and in which direction, does the water push the swimmer?

By the third law, the reaction is equal in magnitude and opposite in direction to the action.
The swimmer's push on the water (action) is 200 N backward.
The water's push on the swimmer (reaction) is therefore 200 N forward.

Answer:200 N, directed forward — which is what propels the swimmer.

Practice this conceptself-check · 4 quick reps

Try it yourself

A book of weight 12 N rests on a table. State the action-reaction pair between the book and the table, and explain why they do not cancel.

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Action and reaction forces act on the same body or different bodies?
2.
How do the magnitudes of an action-reaction pair compare?
3.
How do their directions compare?
4.
Why don't action and reaction cancel out?

Action-reaction pairs act on DIFFERENT bodies

The classic error is to think action and reaction cancel and so nothing moves. They never cancel because they act on two separate bodies. When analysing one body's motion, only the forces ON THAT body matter — its reaction on something else is irrelevant to its own free-body diagram.

Drill 1 more on third law — action and reaction

Concept 4 of 6

Combining forces — the parallelogram law

Intuition

When several forces act at a point, only their VECTOR sum — the resultant — matters for the motion. For two forces the resultant is the diagonal of the parallelogram they span, and its size depends on the angle between them: largest when they are aligned, smallest when they oppose. This is the chapter's main source of HARD questions.

Definition

Two forces $P$ and $Q$ acting at a point with angle $\theta$ between them combine into a resultant $R$ given by the parallelogram law. The resultant is maximum $(P + Q)$ when $\theta = 0^\circ$ and minimum $\lvert P - Q\rvert$ when $\theta = 180^\circ$ . A body is in equilibrium only when the resultant of all forces acting on it is zero.

Magnitude of the resultant of two forces

R = \sqrt{P^2 + Q^2 + 2PQ\cos\theta}

Rmagnitude of the resultant force
P, Qmagnitudes of the two forces
θangle between the two forces

Two forces from a common point combine along the diagonal of the parallelogram they span. The magnitude is R = sqrt(P² + Q² + 2PQ cos θ); the resultant is largest at θ = 0 (P + Q) and smallest at θ = 180 (P - Q).

Worked example

Two forces of 6 N and 8 N act at a point with a 90° angle between them. Find the magnitude of their resultant.

Use $R = \sqrt{P^2 + Q^2 + 2PQ\cos\theta}$ with $P = 6$ , $Q = 8$ , $\theta = 90^\circ$ .
Since $\cos 90^\circ = 0$ , the cross term vanishes: $R = \sqrt{6^2 + 8^2}$ .
$R = \sqrt{36 + 64} = \sqrt{100} = 10\,\text{N}$ .

Answer:10 N.

Practice this conceptself-check · 4 quick reps

Try it yourself

Two equal forces F act at a point and their resultant also has magnitude F. Find the angle between the two forces.

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Resultant of two forces P and Q is maximum at what angle?
2.
Resultant of two forces is minimum at what angle?
3.
Two perpendicular forces 3 N and 4 N have what resultant?
4.
Two equal 10 N forces at 60° give what resultant (to 1 d.p.)?

From the bank · past-year question

Example 4Laws of Motion and ForcesMODERATE

Two forces of 5·0 N each are acting on a point mass. If the angle between the forces is 60°, then the net force acting on the point mass has magnitude close to :

[Q56 · Apr · 2023]

Two equal forces with a resultant equal to each: θ = 120°

NDA 2026 tested two equal forces whose resultant equals one of them. Solving gives cos θ = -1/2, so θ = 120° between them, and each force makes 60° with the resultant. Both statements (60° to resultant, 120° between forces) are correct.

Don't add force magnitudes arithmetically

5 N and 5 N do NOT give 10 N unless they are parallel. At 60° the resultant is √75 ≈ 8.66 N. Always use the parallelogram formula with the angle; only at θ = 0° is the answer the simple sum.

Drill 3 more on combining forces — the parallelogram law

Concept 5 of 6

Mass vs weight

Intuition

Mass is how much matter a body contains — it is the same on Earth, the Moon, or in deep space. Weight is the gravitational force on that mass, W = mg, so it changes with location because g changes. A 60 kg person has the same mass everywhere but weighs about a sixth as much on the Moon.

Definition

Mass is the amount of matter in a body and a measure of its inertia; it is a scalar, measured in kg, and is the same everywhere. Weight is the gravitational force on the body, $W = mg$ ; it is a vector (points down), measured in newtons, and varies with $g$ (location). From $F = ma$ , mass is the constant of proportionality between force and acceleration — weight is not.

Property	Mass	Weight
What it is	Amount of matter / inertia	Gravitational force on the body
Formula	—	W = mg
SI unit	kilogram (kg)	newton (N)
Scalar or vector	Scalar	Vector (downward)
Varies with location?	No — same everywhere	Yes — changes with g NDA 2018 — mass is "the same everywhere"; NDA 2021 — mass is the constant of proportionality in F = ma.

Mass is constant and is the proportionality constant in F = ma; weight = mg varies with g. NDA tests both halves of this distinction.

Practice this conceptself-check · 4 quick reps

Try it yourself

An object has a mass of 60 kg on Earth. What is its mass on the Moon (g_moon ≈ 1.6 m/s²), and what is its weight there?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Does mass change when you go to the Moon?
2.
What is the formula for weight?
3.
Is weight a scalar or a vector?
4.
What is the weight of a 60 kg person on Earth (g = 9.8)?

From the bank · past-year question

Example 5Laws of Motion and ForcesEASY

Which one of the following statements about the mass of a body is correct?

[Q80 · Apr · 2018]

Mass is the constant of proportionality, NOT weight

NDA 2021 asked which is the constant of proportionality between force and acceleration in F = ma. It is MASS, not weight. Weight = mg varies with g; mass is invariant and is what makes a body resist acceleration.

Drill 1 more on mass vs weight

Concept 6 of 6

Rotational inertia — moment of inertia of common bodies

Intuition

Just as mass resists changes in straight-line motion, moment of inertia resists changes in rotation. For the same mass and radius, how that mass is distributed matters: a ring (all mass at the rim) resists spinning more than a disc (mass spread inward), which resists more than a solid sphere. NDA tests this as a direct comparison.

Definition

The moment of inertia $I$ is the rotational analogue of mass — it measures resistance to angular acceleration and depends on how mass is distributed about the axis. For the same mass $M$ and radius $R$ about the central axis: ring $I = MR^2$ , disc $I = \tfrac{1}{2}MR^2$ , solid sphere $I = \tfrac{2}{5}MR^2$ . Rotational kinetic energy is $\tfrac{1}{2}I\omega^2$ , so at the same $\omega$ a larger $I$ means more energy.

Moment of inertia of common bodies (mass M, radius R)

I_{\text{ring}} = MR^2, \quad I_{\text{disc}} = \tfrac{1}{2}MR^2, \quad I_{\text{sphere}} = \tfrac{2}{5}MR^2

Imoment of inertia (kg m²)
Mmass of the body
Rradius about the central axis

Worked example

A solid disc and a solid sphere have the same mass M and the same radius R. Which has the greater moment of inertia about its centre?

Disc: $I_{\text{disc}} = \tfrac{1}{2}MR^2 = 0.5\,MR^2$ .
Sphere: $I_{\text{sphere}} = \tfrac{2}{5}MR^2 = 0.4\,MR^2$ .
Compare the coefficients: $0.5 > 0.4$ .

Answer:The disc — its moment of inertia (0.5 MR²) exceeds the sphere's (0.4 MR²).

Practice this conceptself-check · 4 quick reps

Try it yourself

A thin ring and a thin disc have the same mass and radius and spin at the same angular speed ω. Which has the greater rotational kinetic energy?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Moment of inertia of a ring of mass M, radius R about its centre?
2.
Moment of inertia of a solid disc (mass M, radius R) about its centre?
3.
Moment of inertia of a solid sphere (mass M, radius R)?
4.
What is the rotational analogue of mass?

From the bank · past-year question

Example 6Laws of Motion and ForcesMODERATE

A solid disc and a solid sphere have the same mass and same radius. Which one has the higher moment of inertia about its centre of mass?

[Q64 · Sep · 2019]

Same mass + radius, different I — distribution decides

Ring > disc > solid sphere for moment of inertia at equal M and R, because the ring keeps all its mass at the rim while the sphere packs mass near the axis. The body with mass concentrated farther from the axis always has the larger I.

Drill 1 more on rotational inertia — moment of inertia of common bodies

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (5)

First law — inertia
Condition for the first law (equilibrium of motion)
$\vec{F}_{\text{net}} = 0 \iff \vec{a} = 0$
Second law — F = ma
Newton's second law
$\vec{F} = \frac{d\vec{p}}{dt} = m\vec{a} \quad (\text{constant } m)$
Third law — action and reaction
Newton's third law (force pair)
$\vec{F}_{AB} = -\vec{F}_{BA}$
Combining forces — the parallelogram law
Magnitude of the resultant of two forces
$R = \sqrt{P^2 + Q^2 + 2PQ\cos\theta}$
Rotational inertia — moment of inertia of common bodies
Moment of inertia of common bodies (mass M, radius R)
$I_{\text{ring}} = MR^2, \quad I_{\text{disc}} = \tfrac{1}{2}MR^2, \quad I_{\text{sphere}} = \tfrac{2}{5}MR^2$

Reference tables (1)

Mass vs weight5 rows

Property	Mass	Weight
What it is	Amount of matter / inertia	Gravitational force on the body
Formula	—	W = mg
SI unit	kilogram (kg)	newton (N)
Scalar or vector	Scalar	Vector (downward)
Varies with location?	No — same everywhere	Yes — changes with g NDA 2018 — mass is "the same everywhere"; NDA 2021 — mass is the constant of proportionality in F = ma.

Mass is constant and is the proportionality constant in F = ma; weight = mg varies with g. NDA tests both halves of this distinction.

Watch out for (8)

Constant velocity does NOT mean changing speed
→ First law — inertia
Force is proportional to rate of change of momentum, NOT to momentum itself
→ Second law — F = ma
Watch the units when computing F = ma
→ Second law — F = ma
Action-reaction pairs act on DIFFERENT bodies
→ Third law — action and reaction
Two equal forces with a resultant equal to each: θ = 120°
→ Combining forces — the parallelogram law
Don't add force magnitudes arithmetically
→ Combining forces — the parallelogram law
Mass is the constant of proportionality, NOT weight
→ Mass vs weight
Same mass + radius, different I — distribution decides
→ Rotational inertia — moment of inertia of common bodies

Mastery check — 5 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1Laws of Motion and ForcesEASY

What is the upward force acting on a skydiver of mass 60 kg falling at a uniform speed of 2 m/s?

[Q70 · Apr · 2026]

Example 2Laws of Motion and ForcesEASY

A 5 N force is defined when a mass of 10 kg is accelerated with

[Q80 · Apr · 2022]

Example 3Laws of Motion and ForcesMODERATE

Two identical spring balances

S_1

and

S_2

are connected one after the other and are held vertically as shown in the figure. A mass of 10 kg is hanging from

S_2

. If the readings on

S_1

and

S_2

are

W_1

and

W_2

respectively, then :

[Q51 · Apr · 2023]

Example 4Laws of Motion and ForcesHARD

Two forces of equal magnitude simultaneously act at a point. It is observed that the resultant force is equal in magnitude to the individual forces. Which of the following statements is/are correct? I. The angle between one of the individual forces and the resultant force is

\pi/3

. II. The angle between the individual forces is

2\pi/3

. Select the answer using the code given below.

[Q52 · Apr · 2026]

Example 5Laws of Motion and ForcesMODERATE

Weight and mass of an object are defined with Newton's laws of motion. Which among the following is true?

[Q115 · Sep · 2021]

Drill every past-year question on this subtopic

19 questions from the bank — paginated, with cart and Word-export support.

Drill the 19 questions