Linear Momentum: Study Guide (Physics 1 4.1)

This guide is for the algebra-based Physics 1 course and opens Unit 4, Linear Momentum. Everything here uses algebra and diagrams. You already know two ways to describe a moving object: its velocity (Unit 1) and its kinetic energy (Unit 3). Momentum is a third description. It turns out to be the best tool for short, violent interactions such as crashes, kicks and things breaking apart.

Momentum in everyday speech and in physics

In everyday speech, “momentum” means something like “hard to stop” or “keeps going”. A team on a winning streak “has momentum”. Physics uses the word for one exact quantity:

p = mv

• p is the linear momentum (kg·m/s),
• m is the mass (kg),
• v is the velocity (m/s).

The everyday idea is not far off. A heavy, fast object has a lot of momentum, and you will see in Topic 4.2 that it takes a large push, or a push for a long time, to change it. But the physics definition has two features that everyday speech misses:

1. Momentum depends on both mass and velocity. A slow, massive object can have more momentum than a fast, light one.
2. Momentum has a direction.

Unless a question says otherwise, the single word “momentum” in this course means linear momentum.

Momentum is a vector

Mass is never negative, so multiplying a velocity by a mass does not change its direction. Momentum always points the same way as the velocity.

In one dimension, choose an axis first, for example “+x to the right”. Then

p_x = m v_x

and p_x takes the sign of v_x. A 0.20 kg ball moving at 3.0 m/s to the right has p_x = +0.60 kg·m/s. The same ball moving at 3.0 m/s to the left has p_x = −0.60 kg·m/s. Its speed and its kinetic energy are the same in both cases, but its momentum is not. This is why a ball that bounces straight back has had its momentum changed, even if its speed has not. Topic 4.2 measures that change.

Example values

These are rounded example values to give you a feel for the sizes involved:

Object (example values)	m (kg)	speed (m/s)	size of momentum (kg·m/s)
marble rolling across a table	0.010	2.0	0.020
basketball thrown in a pass	0.60	8.0	4.8
runner	70	5.0	350
car in town traffic	1200	15	18 000

Factors of change

Because p = mv is a simple product, you can predict changes without a full calculation:

• Double the speed, same mass: momentum doubles.
• Triple the mass and halve the speed: momentum changes by 3 × ½ = 1.5.
• Same speed, direction reversed: same size of momentum, opposite sign.

Momentum of a system

In Topic 2.1 you chose systems of more than one object. The total momentum of a system is the vector sum of the momenta of its parts:

p_sys = p₁ + p₂ + p₃ + …

In one dimension, “vector sum” means you add the signed values. Momenta in opposite directions partly or fully cancel. A system can have zero total momentum even though every part of it is moving.

Notice that cart A is heavier but cart B has more momentum, because B is three times faster. Mass alone does not decide which object “has more momentum”.

Topic 4.3 builds on this sum. It links the system momentum to the velocity of the system’s center of mass and shows when the total stays constant.

Momentum in two dimensions

In two dimensions, use components. Each component of momentum is the mass times the matching component of velocity:

p_x = m v_x p_y = m v_y

Then combine them with Pythagoras, as you did for velocity in Topic 1.5.

For example, take +x east, +y north. A 2.0 kg delivery drone flies with v_x = +3.0 m/s and v_y = +4.0 m/s. Its momentum components are p_x = +6.0 kg·m/s and p_y = +8.0 kg·m/s. The size of its momentum is √(6.0² + 8.0²) = 10 kg·m/s, at tan⁻¹(8.0 ÷ 6.0) = 53° north of east. That is exactly the direction of its velocity, as it must be.

Graphs of momentum

You may be asked to sketch, not calculate, how momentum depends on another quantity.

Momentum against velocity. For a fixed mass, p_x = m v_x is a straight line through the origin. Its slope is the mass. Negative velocities give negative momenta, so the line continues into the third quadrant. A heavier object gives a steeper line.

Momentum against time. Because p_x = m v_x and m is constant, a momentum–time graph has the same shape as the velocity–time graph, stretched vertically by a factor m. If v_x is constant, p_x is a horizontal line. If v_x changes steadily, p_x changes steadily too. If the object reverses, p_x crosses zero at the instant it is momentarily at rest. Topic 4.2 shows that the slope of this graph is the net external force.

Collisions and explosions as models

Momentum is most useful for interactions that are short and intense. The course names two models.

A collision is an interaction in which the forces the objects exert on each other are much larger than the net external force on them while the interaction lasts. Picture two lab carts bumping. During a contact lasting a few hundredths of a second, they might push on each other with tens of newtons, while friction from the track is well under a newton. During that brief moment, the external forces hardly matter.

Because we compare only the state just before and just after a collision, we do not need to know how the objects squash and spring back during contact. That is why the object model is allowed: each object can be treated as a single point with a mass and a velocity.

An explosion is an interaction in which forces inside the system push its parts apart. The word covers far more than fireworks. A skater pushing off a partner, a compressed spring launching two carts in opposite directions, and a cannon firing a ball are all explosions in this sense.

Not every interaction fits these models. A sled sliding slowly to a stop on rough snow is not a collision: the only important horizontal force is friction from the ground, which is external, and it acts over a long time.

Worked example 1: total momentum of two carts

Question. Take +x to the right. Cart A (1.2 kg) moves right at 0.50 m/s. Cart B (0.80 kg) moves left at 1.5 m/s. Find the total momentum of the system of both carts. (This is the situation in Figure 1.)

1. Signed velocities: v_A = +0.50 m/s, v_B = −1.5 m/s.
2. Momentum of A: p_A = m_A v_A = 1.2 kg × (+0.50 m/s) = +0.60 kg·m/s.
3. Momentum of B: p_B = m_B v_B = 0.80 kg × (−1.5 m/s) = −1.2 kg·m/s.
4. System: p_sys = p_A + p_B = 0.60 + (−1.2) = −0.60 kg·m/s, that is, 0.60 kg·m/s to the left.

Check. Two tempting shortcuts both fail. Adding sizes, 0.60 + 1.2 = 1.8 kg·m/s, ignores direction. Multiplying the plain mean velocity by the total mass, ½(0.50 − 1.5) × 2.0 = −1.0 kg·m/s, ignores that the carts have different masses. Momentum must be found object by object, then added with signs.

Worked example 2: same momentum, different kinetic energy

Question. A 1500 kg car moves at 12 m/s. A 9000 kg truck moves at 2.0 m/s in the same direction. Compare (a) their momenta and (b) their kinetic energies.

1. Car: p = 1500 kg × 12 m/s = 18 000 kg·m/s.
2. Truck: p = 9000 kg × 2.0 m/s = 18 000 kg·m/s. The momenta are equal.
3. Car: K = ½ × 1500 × 12² = 108 000 J (about 1.1 × 10⁵ J).
4. Truck: K = ½ × 9000 × 2.0² = 18 000 J.
5. Ratio: the car has 108 000 ÷ 18 000 = 6 times the kinetic energy of the truck.

Interpretation. Momentum grows with v, but kinetic energy grows with v². The car is 6 times faster but 6 times lighter, so the factors cancel for momentum. For kinetic energy, the speed factor counts twice (6² = 36) while the mass factor counts once (÷ 6), leaving a factor of 6.

Check. Substituting v = p/m into K = ½mv² gives K = p² ÷ (2m). With equal p, the object with the smaller mass has the larger K: 18 000² ÷ (2 × 1500) = 108 000 J, and 18 000² ÷ (2 × 9000) = 18 000 J. Both match.

Common misconceptions

• “Momentum and kinetic energy are the same idea.” They are not. Momentum is a vector and scales with v; kinetic energy is a scalar and scales with v² (Worked example 2).
• “The heavier object always has more momentum.” Only at the same speed. Speed matters just as much as mass (Figure 1).
• “Total momentum is the sum of the sizes.” Add signed values or vectors. Opposite momenta cancel (Worked example 1).
• “A ball that bounces back at the same speed has the same momentum.” Its momentum has reversed sign, so it has changed.
• “A system with zero momentum must be at rest.” Its parts can move in opposite directions with momenta that cancel.
• “Momentum can be negative, so it can be ‘less than nothing’.” The sign shows direction along your chosen axis, not an amount below zero.
• “Collisions are only between things that crash hard.” The model only needs the internal forces to be much larger than the net external force during the interaction.
• “Explosions need fire.” Any internal push that sends parts of a system apart, such as a released spring, counts.

Where this leads

Momentum on its own is a description. Its power comes from how it changes. Topic 4.2, Change in Momentum and Impulse, links the change in momentum to force and time, and Topic 4.3 shows when the total momentum of a system stays the same. Try the practice questions now, then use the revision notes and the checklist to consolidate. You can also return to the course roadmap.