Conservation of Linear Momentum

students, have you ever watched two billiard balls collide, two skaters push off each other, or a car crash test and wondered what stays the same before and after the impact? 🚗💥 In physics, one of the most powerful ideas for describing collisions is conservation of linear momentum. This lesson will help you understand what momentum is, when it is conserved, and how to use that idea to solve AP Physics C: Mechanics problems.

What Momentum Means

Linear momentum is a measure of how hard it is to stop an object that is moving. It depends on both mass and velocity, so a heavy truck moving slowly can have the same momentum as a small car moving fast. The momentum of one object is

$$\vec{p}=m\vec{v}$$

where $\vec{p}$ is momentum, $m$ is mass, and $\vec{v}$ is velocity.

Because velocity has direction, momentum also has direction. That matters a lot in collisions. If an object moves to the right, its momentum is positive in that direction if we choose right as positive. If it moves left, its momentum is negative.

For a system of objects, the total momentum is the vector sum of the momenta of all the parts:

$$\vec{p}_{\text{total}}=\sum \vec{p}_i=\sum m_i\vec{v}_i$$

This is the quantity that is often conserved in AP Physics C problems.

Big idea

Momentum is useful because it can stay constant even when forces inside a system are large. For example, two ice skaters pushing off each other can change their individual speeds a lot, but the total momentum of the pair can remain the same if external forces are negligible. ❄️

The Law of Conservation of Linear Momentum

The conservation rule says that if the net external impulse on a system is zero, then the system’s total momentum does not change.

Impulse is the change in momentum caused by a force acting over time:

$$\vec{J}=\Delta \vec{p}=\int \vec{F}\,dt$$

If the net external impulse is zero, then

$$\Delta \vec{p}_{\text{total}}=0$$

which means

$$\vec{p}_{\text{total, initial}}=\vec{p}_{\text{total, final}}$$

This is the conservation of linear momentum.

A very important AP Physics C idea is that momentum conservation is not automatic for every situation. It works best when the system is isolated or nearly isolated, meaning the external forces are either absent or their total impulse is negligible over the time interval of interest.

When is momentum conserved?

Momentum is conserved when external forces do not produce a significant net impulse on the system. Common examples include:

Collisions between carts on a low-friction track
Explosions or separations in space
Skaters pushing away from each other on ice, if friction is small
Recoil of a gun and bullet, during the short firing time

A common mistake is to assume momentum is conserved whenever forces are involved. That is not true. A force from outside the system can change the system’s total momentum. For example, a soccer ball rolling on grass slows down because friction from the ground is an external force.

Choosing the Right System

students, one of the most important skills in this topic is deciding what objects belong in the system. The system is the group of objects you analyze together.

If you choose both colliding objects as your system, then the forces between them are internal forces. Internal forces do not change the total momentum of the system because they come in equal and opposite pairs according to Newton’s third law. That is why momentum conservation is so useful in collision problems.

If you choose only one object, then the force from the other object becomes external to that system, so momentum is not conserved for that one-object system during the collision.

Example: two carts

Suppose cart $1$ with mass $m_1$ moves right with velocity $v_{1i}$ and cart $2$ with mass $m_2$ is at rest. If the carts collide on a nearly frictionless track, then the total momentum before and after the collision is

$$m_1v_{1i}+m_2v_{2i}=m_1v_{1f}+m_2v_{2f}$$

If cart $2$ starts from rest, then $v_{2i}=0$, and the equation becomes

$$m_1v_{1i}=m_1v_{1f}+m_2v_{2f}$$

This one equation can be used to solve for an unknown final velocity if enough information is given.

Types of Collisions and Momentum

Momentum conservation applies to all kinds of collisions, whether they are elastic, inelastic, or perfectly inelastic. The difference is what happens to kinetic energy.

Elastic collision

In an elastic collision, both momentum and kinetic energy are conserved:

$$\vec{p}_{\text{initial}}=\vec{p}_{\text{final}}$$

and

$$K_{\text{initial}}=K_{\text{final}}$$

Elastic collisions are idealized, but some collisions, like billiard balls or nearly hard-sphere interactions, can be approximately elastic.

Inelastic collision

In an inelastic collision, momentum is conserved but kinetic energy is not. Some mechanical energy is transformed into heat, sound, or deformation.

Perfectly inelastic collision

In a perfectly inelastic collision, the objects stick together after colliding. Momentum is still conserved, but kinetic energy decreases as much as possible while still satisfying momentum conservation.

If two objects stick together, they share a common final velocity $v_f$, so

$$m_1v_{1i}+m_2v_{2i}=(m_1+m_2)v_f$$

This formula is very common on the AP exam.

Real-world example

When a car crashes into a barrier, the car’s momentum changes rapidly. If you include only the car as the system, the wall is an external object and momentum is not conserved for the car alone. If you include the Earth in the system, the total momentum of the car-Earth system is conserved, but the Earth’s change in velocity is extremely tiny because its mass is so large.

Momentum in Two Dimensions

Sometimes collisions happen at an angle, not just along a straight line. Then momentum conservation must be applied separately in each direction.

For a two-dimensional problem,

$$\sum p_{x,i}=\sum p_{x,f}$$

and

$$\sum p_{y,i}=\sum p_{y,f}$$

This works because momentum is a vector. The $x$- and $y$-components are independent equations.

Example: a puck collision

Imagine a hockey puck moving east collides with a second puck initially at rest, and after the collision they move off at angles. To solve the problem, break each momentum vector into components.

If the first puck has initial momentum only in the $x$-direction, then the initial $y$-momentum is zero. After the collision, the total $y$-momentum must still be zero, so one puck’s upward component must be balanced by the other puck’s downward component.

This kind of problem often combines geometry and momentum equations. A good strategy is to draw a diagram, label angles, and write one conservation equation for each axis.

How to Solve Momentum Conservation Problems

Here is a reliable AP Physics C procedure, students ✅

Choose the system. Decide which objects are included.
Check whether external impulse is negligible. If not, momentum may not be conserved.
Choose a coordinate system. Pick positive directions for each axis.
Write momentum conservation equations. Use components if needed.
Substitute known values. Keep signs consistent.
Solve algebraically before plugging in numbers when possible.
Check units and whether the answer makes physical sense.

Example with numbers

A $2.0\,\text{kg}$ cart moving at $3.0\,\text{m/s}$ collides and sticks to a $1.0\,\text{kg}$ cart at rest. What is the final velocity?

Use momentum conservation:

$$m_1v_{1i}+m_2v_{2i}=(m_1+m_2)v_f$$

Substitute:

$$\left(2.0\right)\left(3.0\right)+\left(1.0\right)\left(0\right)=\left(3.0\right)v_f$$

$$6.0=3.0v_f$$

and

$$v_f=2.0\,\text{m/s}$$

The stuck-together carts move in the original direction of the first cart.

Connection to Newton’s Laws and Impulse

Conservation of momentum is closely connected to Newton’s laws. Newton’s third law explains why internal forces in a system cancel in pairs when you look at the total momentum of the whole system. Newton’s second law gives the impulse-momentum relationship:

$$\vec{F}_{\text{net}}=\frac{d\vec{p}}{dt}$$

If the net external force on a system is zero, then

$$\frac{d\vec{p}_{\text{total}}}{dt}=0$$

which means total momentum stays constant.

This connection is important because AP Physics C often asks you to reason both mathematically and conceptually. You should be able to explain why momentum is conserved, not just plug numbers into a formula.

Conclusion

Conservation of linear momentum is one of the most useful ideas in mechanics because it lets you analyze collisions and separations even when the forces are complicated and change quickly. The key is to define a system, check for external impulse, and apply

$$\vec{p}_{\text{initial}}=\vec{p}_{\text{final}}$$

for the whole system when the conditions are right. This topic connects directly to momentum, impulse, Newton’s laws, and energy. students, if you can identify the system, conserve momentum in each direction, and interpret the result physically, you are well prepared for AP Physics C: Mechanics problems involving collisions and recoil. 🚀

Study Notes

Momentum is defined as $\vec{p}=m\vec{v}$.
Total momentum is the vector sum $\vec{p}_{\text{total}}=\sum m_i\vec{v}_i$.
Momentum is conserved when the net external impulse is zero.
Impulse equals change in momentum: $\vec{J}=\Delta \vec{p}=\int \vec{F}\,dt$.
For an isolated system, $\vec{p}_{\text{initial}}=\vec{p}_{\text{final}}$.
Internal forces do not change the total momentum of the full system.
In one dimension, use signs carefully; in two dimensions, conserve momentum separately in $x$ and $y$.
Momentum is conserved in elastic, inelastic, and perfectly inelastic collisions.
Kinetic energy is conserved only in elastic collisions.
Perfectly inelastic collisions satisfy $m_1v_{1i}+m_2v_{2i}=(m_1+m_2)v_f$.
A good problem-solving strategy is: choose system, check external impulse, write conservation equations, and verify the answer.
Momentum conservation is a central AP Physics C topic because it connects forces, motion, and collision analysis.