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Momentum
\( \vec{p} = m\vec{v} \)

Explore linear momentum, the impulse-momentum theorem, and the conservation of momentum to analyze collisions and interactions in physics.

AP Physics 1

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. A heavy truck moving slowly can have the same momentum as a light car moving quickly.

Momentum Is a Vector

x y A +p B −p Positive direction Negative direction

Units and Dimensions

SI Unit
\( \text{kg} \cdot \text{m/s} \)
There is no special name for the unit of momentum.
Dimensions
\( [M][L][T^{-1}] \)
Mass times length per time.
Typical Values
Baseball pitch: ~\( 6 \text{ kg} \cdot \text{m/s} \)
Car at 60 mph: ~\( 40,000 \text{ kg} \cdot \text{m/s} \)

The Impulse-Momentum Theorem

Newton's Second Law can be rewritten in terms of momentum. The net force on an object equals the rate of change of its momentum:

t F
\( J = F \cdot \Delta t \)
\( t_1 \)
\( t_2 \)
\( F \)
 Area = Impulse

Conservation of Momentum

When no net external force acts on a system of objects, the total momentum of the system remains constant. This is one of the most powerful principles in physics.

Visualizing a Collision

Before
\( m_1 \)
\( v_{1i} \)
\( m_2 \)
 at rest → After
\( m_1 \)
\( v_{1f} \)
\( m_2 \)
\( v_{2f} \)
\( m_1 v_{1i} + m_2(0) = m_1 v_{1f} + m_2 v_{2f} \)
Total momentum is conserved

Additional worked example

The following is a separate example from the long-form lesson. Its values may differ from the example already on this slide.

A 3.0 kg cart (Cart A) traveling east at 5.0 m/s collides head-on with a 2.0 kg cart (Cart B) traveling west at 4.0 m/s. The carts stick together after the collision. Find the velocity of the combined carts after the collision and determine how much kinetic energy is lost.

Show answer and explanation

Step 1 — Define the system and sign convention: The system consists of Cart A and Cart B. Choose east as positive. Therefore: v_(A,i) = +5.0 m/s and v_(B,i) = −4.0 m/s (west is negative). Because the carts stick together, this is a perfectly inelastic collision. Step 2 — Apply conservation of momentum: m_(A)v_(A,i) + m_(B)v_(B,i) = (m_(A) + m_(B))v_(f). Substituting: (3.0)(+5.0) + (2.0)(−4.0) = (3.0 + 2.0)v_(f). Step 3 — Solve for the final velocity: 15.0 + (−8.0) = 5.0 × v_(f) → 7.0 = 5.0 v_(f) → v_(f) = +1.4 m/s. The positive sign means the combined mass moves east. v_f = +1.4 m/s (east) Step 4 — Calculate kinetic energy before and after: KE_(i) = ½(3.0)(5.0)² + ½(2.0)(4.0)² = 37.5 + 16.0 = 53.5 J. KE_(f) = ½(5.0)(1.4)² = 4.9 J. Step 5 — Determine kinetic energy lost: ΔKE = KE_(f) − KE_(i) = 4.9 − 53.5 = −48.6 J. About 91% of the kinetic energy is converted to thermal energy, sound, and deformation during the collision. This large loss is characteristic of perfectly inelastic collisions. ΔKE = −48.6 J (lost to internal energy)

Carts of 3 kg at +5 m/s and 2 kg at −4 m/s couple at +1.4 m/s. Momentum remains +7 kg·m/s; kinetic energy decreases from 53.5 to 4.9 J.
Carts of 3 kg at +5 m/s and 2 kg at −4 m/s couple at +1.4 m/s. Momentum remains +7 kg·m/s; kinetic energy decreases from 53.5 to 4.9 J.

Elastic vs. Inelastic Collisions

Elastic Inelastic Perfectly Inelastic
Momentum Conserved Conserved Conserved
Kinetic Energy Conserved NOT conserved NOT conserved
Objects After Bounce apart May deform Stick together
Example Billiard balls Car crash (bounce) Clay balls merging

How to Solve Momentum Problems

1
Define your system. Identify all objects involved and check if external forces are negligible.
2
Choose a direction as positive. Assign signs to all velocities consistently.
3
Write conservation of momentum: \( m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f} \)
4
Identify the collision type. If perfectly inelastic, set \( v_{1f} = v_{2f} \). If elastic, add a KE equation.
5
Solve for unknowns and verify that the direction of each velocity makes physical sense.

Example 1: Calculating Momentum

A \( 0.145 \text{ kg} \) baseball is pitched at \( 40 \text{ m/s} \). What is its momentum?

Example 2: Impulse on a Ball

A \( 0.5 \text{ kg} \) ball moving at \( 6 \text{ m/s} \) to the right hits a wall and bounces back at \( 4 \text{ m/s} \) to the left. Find the impulse.

Example 3: Perfectly Inelastic Collision

A \( 2 \text{ kg} \) cart moving at \( 3 \text{ m/s} \) collides with a \( 1 \text{ kg} \) stationary cart. They stick together. Find the final velocity.

Why Direction Matters

0
\( 3 \text{ kg} \)
\( +4 \text{ m/s} \)
\( p = +12 \)
\( 4 \text{ kg} \)
\( -3 \text{ m/s} \)
\( p = -12 \)

Systems and External Forces

Internal Forces
Forces between objects in the system (e.g., collision forces). These are equal and opposite (Newton's 3rd Law) and cancel out.
External Forces
Forces from outside the system (e.g., friction, gravity, applied pushes). These change the total momentum.

Common Mistakes

✕ Forgetting Signs
Treating all velocities as positive. If one object moves left, its velocity must be negative.
✕ Confusing p and KE
Momentum is \( mv \) (vector), not \( \frac{1}{2}mv^2 \) (scalar). They are conserved under different conditions.
✕ Wrong System
Applying conservation to a single object instead of the entire system. Both objects must be included.

Key Formulas Summary

Concept Formula When to Use
Momentum \( \vec{p} = m\vec{v} \) Any moving object
Impulse \( \vec{J} = \vec{F}\Delta t = \Delta \vec{p} \) Force over time
Conservation \( \Sigma \vec{p}_i = \Sigma \vec{p}_f \) No net external force
Perfectly Inelastic \( m_1v_{1i} + m_2v_{2i} = (m_1+m_2)v_f \) Objects stick together
Elastic (extra eq.) \( \frac{1}{2}m_1v_{1i}^2 + \frac{1}{2}m_2v_{2i}^2 = \frac{1}{2}m_1v_{1f}^2 + \frac{1}{2}m_2v_{2f}^2 \) KE also conserved

Question 1 of 5

A \( 4 \text{ kg} \) object moves at \( 5 \text{ m/s} \). What is its momentum?

Question 2 of 5

A \( 2 \text{ kg} \) ball moving at \( 3 \text{ m/s} \) to the right collides with a \( 2 \text{ kg} \) ball at rest. They stick together. What is the final velocity?

Question 3 of 5

Which quantity is always conserved during any collision (assuming no net external force)?

Question 4 of 5

A force of \( 100 \text{ N} \) acts on an object for \( 0.02 \text{ s} \). What impulse does the object receive?

Question 5 of 5

A \( 1500 \text{ kg} \) car and a \( 3000 \text{ kg} \) truck are moving toward each other. The car moves at \( +20 \text{ m/s} \) and the truck at \( -10 \text{ m/s} \). What is the total momentum of the system?