CDS Physics · Laws of Motion and Forces

Newton's Laws and Momentum

What happens when the net force is zero, how a net force changes momentum, and when the total momentum of a system stays the same.

All Laws of Motion and Forces notes

Why this matters

Nine CDS questions from 2017 (II) to 2026 (II), three of them in the two 2026 papers. Most test one law in words. The two sums ask for the momentum and kinetic energy of a moving body, and for its final speed after a force acts for a few seconds.

Concept 1 of 3: Zero net force: inertia and equilibrium

A body left alone keeps doing what it was doing: at rest it stays at rest, moving it keeps moving in a straight line at the same speed. That is the first law, the law of inertia. When the forces on a body cancel, it is in equilibrium. Whether that equilibrium survives a small push depends on what happens to the centre of gravity when you tilt the body.

Definition

The first law and equilibrium:

First law (inertia): with zero net force, a body stays at rest or moves with constant velocity (same speed, same direction).
On a frictionless horizontal surface, the weight is balanced by the normal reaction, so a moving body keeps a constant velocity. These two forces act on the same body, so they are not a third-law pair.
Equilibrium: net force zero and net moment (turning effect) zero.
Stable: a small tilt raises the centre of gravity, and the body returns (a cone on its base).
Unstable: a small tilt lowers the centre of gravity, and the body falls further (a rod or cone balanced on its tip or end).
Neutral: the centre of gravity stays at the same height (a ball on a flat floor).
A rope holding a weight at its middle: each half pulls up with $T\sin\theta$ , so $2T\sin\theta = mg$ . As the rope is pulled straight, $\theta \to 0$ and $T$ grows without limit: no finite tension can make it perfectly straight.

Situation	What happens
Object sliding on a frictionless horizontal surface	Moves with constant velocity (first law)Q CDS 2017 (II): with the forces balanced and no friction, the object moves with constant velocity.
Rod balanced on one end	Unstable equilibriumQ CDS 2026 (II): a rod balanced vertically on one end is in unstable equilibrium.
Cone resting on its flat base	Stable equilibrium
Ball on a level floor	Neutral equilibrium
Rope with a weight at its middle, pulled straight	Needs an infinitely large tensionQ CDS 2018 (II): the tension needed to make the rope perfectly straight is infinitely large.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2026 · CDS (II) 2026 — General Knowledge · Q56Easy

Example 1 · Laws of Motion and Forces · Newton's Laws and Momentum

A rod is balanced vertically on one of its ends. Which one among the following is the correct nature of its state of equilibrium ?

Weight and normal reaction are not an action-reaction pair

An action-reaction pair acts on two different bodies. The weight and the normal reaction both act on the same body, so when they cancel, that is equilibrium (first law), not the third law.

Zero net force does not mean at rest

With zero net force a body moves with constant velocity, which may be zero or may not. A body already moving keeps moving.

Tilt test for equilibrium

Ask what a small tilt does to the centre of gravity: up means stable, down means unstable, no change means neutral. A rod on its end is unstable because its centre of gravity is as high as it can be.

Drill 2 more on zero net force: inertia and equilibrium

Concept 2 of 3: Second law: force changes momentum

Momentum is how much motion a body carries: its mass times its velocity. A loaded truck and a bicycle at the same speed have very different momentum. The second law says a net force changes momentum, and the rate of that change equals the force. For a body whose mass does not change, that is the familiar force equals mass times acceleration.

Definition

The second law and momentum:

Momentum $p = mv$ , unit kg m/s (the same as N s).
Second law: the rate of change of momentum equals the applied force, $F = \dfrac{\Delta p}{\Delta t}$ . For constant mass, $F = ma$ .
So a force $F$ acting for time $t$ changes the velocity by $\Delta v = \dfrac{F t}{m}$ .
Impulse = force × time = change in momentum.
Kinetic energy $= \tfrac{1}{2}mv^{2} = \dfrac{p^{2}}{2m}$ .
Convert km/h to m/s by multiplying by $\tfrac{5}{18}$ , and back by $\tfrac{18}{5}$ .

Second law of motion

p = mv \qquad F = \dfrac{\Delta p}{\Delta t} = ma \qquad \Delta v = \dfrac{Ft}{m} \qquad K = \tfrac{1}{2}mv^{2} = \dfrac{p^{2}}{2m}

pmomentum
Fnet applied force
ttime for which the force acts
Kkinetic energy

Worked example

A car of mass 1200 kg is moving at 36 km/h. A forward force of 600 N acts on it for 20 s. Find its final speed in km/h.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2026 · CDS (II) 2026 — General Knowledge · Q53Moderate

Example 2 · Laws of Motion and Forces · Newton's Laws and Momentum

A car of mass 800 kg is travelling at a speed of 40 km/h on a straight road. A force of 1000 N is applied on it in the direction of its velocity for 12 s. Which one among the following is the final speed of the car ?

Force changes momentum, not displacement or acceleration

The second law links force to the rate of change of momentum. Statements about the rate of change of displacement (that is velocity) or of acceleration are distractors. For angular momentum the matching cause is a torque, not a force.

Change the speed into m/s before using F = ma

With the force in newtons and the mass in kg, the acceleration comes out in m/s². Convert km/h to m/s first (× 5/18), work out the gain, then convert back (× 18/5) if the answer is wanted in km/h.

Drill 2 more on second law: force changes momentum

Concept 3 of 3: Conservation of momentum

If nothing outside a group of bodies pushes on it, the total momentum of the group cannot change. When a gun fires, the bullet goes forward and the gun kicks back so that the total stays zero. This is also the only way to move on perfectly smooth ground: throw something one way, and you move the other way.

Definition

The rule and its limits:

Total momentum is conserved when the net external force is zero. It is not conserved "always".
For two bodies: $m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$ .
Recoil: starting from rest, $m_1v_1 = -m_2v_2$ , so the lighter body moves faster.
On a frictionless floor you cannot walk, run, jump or roll anywhere, because each needs a push from friction. You can move only by throwing something out: even sneezing or blowing air works.
Kinetic energy is not always conserved: it is lost in inelastic collisions and to friction, though momentum is still conserved in the collision.

Conservation of momentum

m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2 \qquad \text{recoil from rest: } m_1v_1 = -m_2v_2

m₁, m₂masses of the two bodies
u, vvelocities before and after

Worked example

A gun of mass 4 kg fires a bullet of mass 20 g at 400 m/s. Find the recoil speed of the gun.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2025 · CDS (II) 2025 — General Knowledge · Q23Easy

Example 3 · Laws of Motion and Forces · Newton's Laws and Momentum

The linear momentum of a particle is conserved if :

Watch the word 'always'

Momentum is conserved only when the net external force is zero, and kinetic energy is conserved only in elastic collisions. A statement saying either is always conserved is wrong.

Running and rolling need friction

On frictionless ground, running, jumping and rolling all fail because there is nothing to push against. Throwing out mass (sneezing, blowing, throwing a shoe) is the only way to move.

Torque is not the condition for linear momentum

Zero net force conserves linear momentum. Zero net torque conserves angular momentum. Do not swap them.

Drill 2 more on conservation of momentum

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 (2)

Second law: force changes momentum
Second law of motion
$p = mv \qquad F = \dfrac{\Delta p}{\Delta t} = ma \qquad \Delta v = \dfrac{Ft}{m} \qquad K = \tfrac{1}{2}mv^{2} = \dfrac{p^{2}}{2m}$
Conservation of momentum
Conservation of momentum
$m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2 \qquad \text{recoil from rest: } m_1v_1 = -m_2v_2$

Reference tables (1)

Zero net force: inertia and equilibrium5 rows

Situation	What happens
Object sliding on a frictionless horizontal surface	Moves with constant velocity (first law)Q CDS 2017 (II): with the forces balanced and no friction, the object moves with constant velocity.
Rod balanced on one end	Unstable equilibriumQ CDS 2026 (II): a rod balanced vertically on one end is in unstable equilibrium.
Cone resting on its flat base	Stable equilibrium
Ball on a level floor	Neutral equilibrium
Rope with a weight at its middle, pulled straight	Needs an infinitely large tensionQ CDS 2018 (II): the tension needed to make the rope perfectly straight is infinitely large.

Watch out for (8)

Weight and normal reaction are not an action-reaction pair
→ Zero net force: inertia and equilibrium
Zero net force does not mean at rest
→ Zero net force: inertia and equilibrium
Tilt test for equilibrium
→ Zero net force: inertia and equilibrium
Force changes momentum, not displacement or acceleration
→ Second law: force changes momentum
Change the speed into m/s before using F = ma
→ Second law: force changes momentum
Watch the word 'always'
→ Conservation of momentum
Running and rolling need friction
→ Conservation of momentum
Torque is not the condition for linear momentum
→ Conservation of momentum

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