Conservation of Linear Momentum

Learning Goals

After this lesson, you should be able to:

• define a system and distinguish internal from external interactions;
• decide whether total momentum or one momentum component is conserved;
• apply momentum conservation to sticking collisions, separations, and recoil;
• solve two-dimensional events with vector components;
• connect total momentum to center-of-mass velocity;
• include an external impulse when momentum is not conserved; and
• evaluate collision data with uncertainty.

1. The Conservation Statement

The total momentum of a selected system is the vector sum

The impulse–momentum theorem for the whole system is

If the net external impulse during the interval is zero or negligible,

This equality applies component by component and must use velocities measured in one reference frame.

2. Why Internal Forces Do Not Change Total Momentum

During an interaction, object A exerts a force on B while B exerts an equal and opposite force on A. Acting over the same time interval, these forces produce equal and opposite internal impulses:

One object's momentum gain therefore equals the other's momentum loss. Internal forces may be large and may deform, heat, or separate the objects, but their vector contributions cancel in the total-system accounting. Only net impulse transferred across the system boundary changes total momentum.

4. System Boundary and External Impulse

Before writing a conservation equation, draw or name the system boundary. The same interaction can be internal for one choice and external for another.

“No external force” is sufficient but stronger than necessary. The relevant condition is zero or negligible net external impulse over the analyzed interval.

5. A Reliable Problem-Solving Workflow

1. Choose the objects in the system and the before/after interval.
2. Select coordinate directions and use one reference frame.
3. List important external forces and estimate their net impulse.
4. Conserve only the momentum components with negligible external impulse.
5. Write every initial and final object momentum with signs or components.
6. Solve symbolically before substituting values.
7. Check units, direction, and whether the result is physically plausible.

If external impulse is not negligible, do not abandon momentum accounting; include it using \(\vec p_f=\vec p_i+\vec J_{\mathrm{ext}}\).

6. Worked Example: Sticking Collision in One Dimension

Question: A \(2.0\,\mathrm{kg}\) cart moves right at \(4.0\,\mathrm{m/s}\). A \(3.0\,\mathrm{kg}\) cart moves left at \(1.0\,\mathrm{m/s}\). They stick on a nearly frictionless track. Find their common velocity.

Take right as positive. The carts form the system, and the track's external impulse during the short collision is negligible.

Answer: The joined carts move right at \(1.0\,\mathrm{m/s}\). Their kinetic energy is not conserved merely because their momentum is.

7. Explosion and Recoil from Rest

Suppose a stationary object separates into a \(4.0\,\mathrm{kg}\) piece and a \(1.0\,\mathrm{kg}\) piece. If the smaller piece moves right at \(8.0\,\mathrm{m/s}\), the initial total momentum is zero:

The larger piece recoils left. Equal and opposite momenta do not mean equal speeds: the more massive object moves more slowly. Kinetic energy can increase because stored chemical, elastic, or other internal energy is transformed during separation.

8. Center-of-Mass Velocity

For total system mass \(M=\sum_i m_i\), the center-of-mass velocity is

If total mass stays in the chosen system and net external impulse is zero, total momentum is constant, so \(\vec v_{\mathrm{cm}}\) is constant. Objects may collide, stick, bounce, or explode around the center of mass without changing its uniform motion.

In the sticking-cart example, \(p_{\mathrm{sys}}=5.0\,\mathrm{kg\,m/s}\) and \(M=5.0\,\mathrm{kg}\), so \(v_{\mathrm{cm}}=+1.0\,\mathrm{m/s}\) both before and after the collision.

9. Conservation in Two Dimensions

Vector conservation becomes two independent scalar equations:

When both external-impulse components are negligible, conserve both. Keep components signed until the end; only then use the Pythagorean theorem and an angle to describe an unknown vector.

10. Worked Example: Two-Dimensional Collision

Question: A \(0.50\,\mathrm{kg}\) puck moving east at \(6.0\,\mathrm{m/s}\) strikes a stationary \(1.0\,\mathrm{kg}\) puck. Afterward, the first puck moves north at \(2.0\,\mathrm{m/s}\). Find the second puck's velocity on a frictionless surface.

Answer: Its speed is \(\sqrt{10}=3.16\,\mathrm{m/s}\), directed \(18.4^\circ\) south of east.

11. Partial Conservation by Component

A system can conserve momentum along one axis without conserving it along another. For a projectile during flight, gravity supplies vertical external impulse, so the projectile's vertical momentum changes. With negligible air resistance, horizontal external impulse is zero, so its horizontal momentum remains constant.

For a very brief collision on a horizontal surface, gravity and the normal force largely cancel vertically, while friction may deliver so little horizontal impulse that horizontal momentum is approximately conserved. State the approximation rather than silently ignoring forces.

12. Momentum and Kinetic Energy Answer Different Questions

Momentum conservation alone supplies one equation per spatial dimension. Do not add a kinetic-energy conservation equation unless the event is stated or justified to be elastic.

13. When External Impulse Cannot Be Ignored

A \(2.0\,\mathrm{kg}\) cart system initially has momentum \(+6.0\,\mathrm{kg\,m/s}\). An external braking force delivers impulse \(-2.5\,\mathrm{N\,s}\). Then

Answer: The final momentum is \(+3.5\,\mathrm{kg\,m/s}\), not \(+6.0\,\mathrm{kg\,m/s}\). Momentum was transferred from the cart system to its surroundings.

14. Investigation: Test Momentum Conservation

Use two low-friction carts, a balance, photogates or video analysis, and interchangeable bumpers.

1. Measure both cart masses and choose the positive track direction.
2. Record each cart's velocity immediately before and after a collision.
3. Calculate \(p_i=m_1v_{1i}+m_2v_{2i}\) and \(p_f=m_1v_{1f}+m_2v_{2f}\).
4. Compare \(p_i\) and \(p_f\) using uncertainty rather than exact equality.
5. Calculate \(v_{\mathrm{cm}}=p_{\mathrm{sys}}/(m_1+m_2)\) before and after.
6. Repeat with magnetic, spring, and sticking bumpers; compare momentum and kinetic-energy changes separately.
7. Discuss track tilt, rolling resistance, sensor timing, and velocity-fit intervals as possible external or measurement effects.

15. Common Traps and Key Takeaways

• Define the system first; internal and external depend on that choice.
• Conserve vectors, not speed totals.
• Use signed velocities in one dimension and components in two dimensions.
• A collision does not automatically guarantee conservation; assess external impulse.
• Zero total momentum does not mean every object is stationary.
• Equal and opposite momenta do not imply equal speeds.
• Momentum can be conserved when kinetic energy is not.
• With nonzero external impulse, use \(\vec p_f=\vec p_i+\vec J_{\mathrm{ext}}\).