Momentum and Impulse: Physics Formulas Explained

Momentum and impulse are two of the most important concepts in classical mechanics. They describe how objects move and how forces change that motion over time. From car crashes to rocket launches, understanding momentum helps us analyse and predict the outcomes of collisions, explosions, and everyday interactions between objects.

In this guide, we will explore the momentum formula, the impulse formula, and the conservation of momentum. We will work through real-world examples including vehicle collisions, billiard balls, and rocket propulsion, giving you a thorough understanding of these foundational physics concepts.

What Is Momentum?

Momentum is a measure of how difficult it is to stop a moving object. It depends on both the object's mass and its velocity. The momentum formula is:

Where is momentum, is mass (in kilograms), and is velocity (in metres per second). Momentum is a vector quantity, meaning it has both magnitude and direction. The SI unit of momentum is kg·m/s.

A heavy lorry moving slowly can have the same momentum as a light car moving quickly. For example, a 20,000 kg lorry travelling at 2 m/s has a momentum of 40,000 kg·m/s, which is the same as a 1,000 kg car travelling at 40 m/s. Both would be equally difficult to bring to a stop.

What Is Impulse?

Impulse describes the effect of a force acting over a period of time. It is defined as:

Where is impulse, is the average force (in newtons), and is the time interval over which the force acts (in seconds). The SI unit of impulse is N·s, which is equivalent to kg·m/s.

The crucial connection between impulse and momentum is captured by the impulse-momentum theorem:

This tells us that the impulse applied to an object equals the change in its momentum. A large force over a short time produces the same change in momentum as a small force over a long time. This principle is the reason airbags and crumple zones save lives, as we will discuss later.

Conservation of Momentum

One of the most powerful principles in physics is the law of conservation of momentum. It states that in any closed system (one with no external forces), the total momentum before an interaction equals the total momentum after:

This law applies to all collisions, explosions, and interactions between objects. It is a direct consequence of Newton's third law: when two objects interact, they exert equal and opposite forces on each other for the same duration, so the impulses are equal and opposite, meaning the total change in momentum is zero.

Elastic vs Inelastic Collisions

Collisions are classified based on whether kinetic energy is conserved in addition to momentum.

Elastic Collisions

In a perfectly elastic collision, both momentum and kinetic energy are conserved. The objects bounce off each other without any energy being lost to deformation, heat, or sound. True elastic collisions are rare in everyday life but are a good approximation for collisions between hard objects like billiard balls or between atoms and molecules.

For an elastic collision between two objects:

Inelastic Collisions

In an inelastic collision, momentum is conserved but kinetic energy is not. Some kinetic energy is converted into other forms such as heat, sound, or deformation. Most real-world collisions are inelastic.

A perfectly inelastic collision is one where the objects stick together after the collision. This represents the maximum possible loss of kinetic energy while still conserving momentum:

Worked Examples

Example 1: Car Collision (Perfectly Inelastic)

A 1,500 kg car travelling east at 20 m/s collides with a stationary 2,500 kg van. The two vehicles lock together after the collision. What is their combined velocity after the impact?

Using conservation of momentum:

The combined wreckage moves east at 7.5 m/s. We can also calculate the kinetic energy lost:

The collision dissipated 187,500 J of energy (62.5%) as heat, sound, and deformation of the vehicles. This is typical of real-world vehicle collisions.

Example 2: Billiard Balls (Elastic Collision)

A billiard ball of mass 0.17 kg travels at 4 m/s and strikes an identical stationary ball head-on in a perfectly elastic collision. What are the velocities of both balls after the collision?

For an elastic head-on collision between equal masses, there is an elegant result: the moving ball stops completely and the stationary ball moves off with the original velocity. Let us verify this using both conservation laws.

Conservation of momentum:

Conservation of kinetic energy:

Solving these simultaneous equations gives m/s and m/s. The first ball stops and the second ball moves off at 4 m/s in the same direction. You can observe this elegant momentum transfer every time you play pool or snooker.

Example 3: Rocket Propulsion

A 5,000 kg rocket in deep space (no external forces) ejects 50 kg of exhaust gas at a velocity of 3,000 m/s relative to the rocket. What is the rocket's change in velocity?

The initial total momentum is zero (everything is at rest). By conservation of momentum:

Note that the rocket mass is now 4,950 kg (after ejecting 50 kg of fuel), and the gas velocity is negative because it moves in the opposite direction:

The rocket gains a velocity of approximately 30.3 m/s in the forward direction. This is the fundamental principle behind all rocket propulsion: by continuously ejecting mass at high speed in one direction, the rocket accelerates in the opposite direction.

Example 4: Impulse and Force

A 0.145 kg cricket ball travelling at 35 m/s is hit by a bat and travels back at 45 m/s. The ball is in contact with the bat for 0.002 seconds. What is the average force exerted by the bat on the ball?

First, calculate the change in momentum (taking the initial direction as positive):

Now use the impulse-momentum theorem:

The average force is 5,800 N (about 580 times the weight of the ball) directed back towards the bowler. The negative sign indicates the force is in the opposite direction to the ball's original motion.

The Impulse-Momentum Theorem in Detail

The impulse-momentum theorem is derived directly from Newton's second law. Starting with and recognising that acceleration is the rate of change of velocity:

Rearranging gives:

This means that a given change in momentum can be achieved by a large force over a short time or a small force over a long time. This insight has profound practical applications.

Real-World Applications

Airbags and Crumple Zones

When a car crashes, the occupants must decelerate from their travelling speed to zero. The change in momentum is fixed (determined by the occupant's mass and initial speed). Airbags and crumple zones increase the time over which this deceleration occurs, which reduces the force experienced by the occupants.

Consider a 70 kg person in a car travelling at 15 m/s (about 54 km/h). Their momentum is kg·m/s. Without safety features, they might stop in 0.05 s, experiencing a force of N. With an airbag extending the stopping time to 0.3 s, the force drops to N. That is a six-fold reduction.

Sports

Athletes instinctively apply the impulse-momentum theorem. A cricket fielder pulls their hands back when catching a fast ball, increasing the stopping time and reducing the force on their hands. A boxer rolls with a punch rather than bracing against it, extending the contact time and reducing the impact force.

In tennis and golf, players follow through with their swing. This maximises the contact time between the racket or club and the ball, delivering a greater impulse and therefore a greater change in momentum, resulting in a faster shot.

Space Exploration

Every spacecraft manoeuvre relies on conservation of momentum. The Tsiolkovsky rocket equation, which determines the maximum velocity a rocket can achieve, is derived from the momentum principle applied continuously as fuel is burned and expelled:

Where is the exhaust velocity, is the initial mass (with fuel), and is the final mass (without fuel). This equation governs the design of every rocket ever built.

Units Summary

Keeping track of units is essential when solving momentum and impulse problems. Here is a quick reference:

• Momentum: kg·m/s (kilogram metres per second)
• Impulse: N·s (newton seconds), which is equivalent to kg·m/s
• Force: N (newtons), where 1 N = 1 kg·m/s²
• Mass: kg (kilograms)
• Velocity: m/s (metres per second)
• Time: s (seconds)

The equivalence of N·s and kg·m/s follows from the definition of a newton: , so .

Frequently Asked Questions

What is the difference between momentum and kinetic energy?

Momentum () is a vector quantity, meaning it has direction. Kinetic energy () is a scalar quantity with no direction. Momentum is always conserved in collisions, but kinetic energy is only conserved in elastic collisions. The two quantities also scale differently with velocity: doubling the speed doubles the momentum but quadruples the kinetic energy.

Can momentum be negative?

Yes. Since momentum is a vector, its sign indicates direction. If you define rightward as positive, then an object moving leftward has negative momentum. This is crucial when solving collision problems, as you must be consistent with your sign convention.

Why do airbags save lives?

Airbags increase the time over which a person decelerates during a crash. Since the change in momentum is fixed (determined by mass and speed), increasing the time reduces the force, according to . A smaller force means less risk of injury. The same principle applies to crumple zones, padded dashboards, and seatbelt pre-tensioners.

What is the conservation of momentum used for?

Conservation of momentum is used to analyse collisions (car crashes, particle physics experiments), explosions (fireworks, bullet firing from a gun), rocket propulsion, and any interaction where objects exchange forces. It is one of the most fundamental laws in physics and holds true in every situation where no external net force acts on the system.

How does rocket propulsion work in terms of momentum?

A rocket works by ejecting mass (hot exhaust gas) at high speed in one direction. By conservation of momentum, the rocket must gain momentum in the opposite direction. The faster the exhaust is ejected and the more mass is expelled, the greater the rocket's change in velocity. This works even in the vacuum of space because the rocket pushes against its own exhaust, not against the air.