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Momentum
Conservation in Collisions

Master the law that governs every collision: from billiard balls to car crashes. Learn to apply \( \vec{p}_{\text{before}} = \vec{p}_{\text{after}} \) to solve real problems.

High School Physics

What Is Momentum?

Momentum is a measure of how hard it is to stop a moving object. It depends on both mass and velocity:

Large Mass
A slow freight train has enormous momentum because \( m \) is huge.
High Velocity
A small bullet has large momentum because \( v \) is very high.

Impulse: Changing Momentum

A force applied over time changes an object's momentum. This quantity is called impulse:

This theorem connects Newton's Second Law to momentum. In fact, Newton originally wrote his second law as \( \vec{F} = \frac{d\vec{p}}{dt} \).

Why Momentum Is Conserved

A \( F_{A \to B} \) B \( F_{B \to A} \) Newton's 3rd Law \( F_{A \to B} = -F_{B \to A} \) Internal forces cancel Total momentum unchanged

The Conservation of Momentum

1
Closed system: no mass enters or leaves the system.
2
Isolated system: the net external force on the system is zero (or negligible during the collision).
3
Vector equation: apply separately to each component (\( x \) and \( y \)) if needed.

Before and After a Collision

BEFORE \( m_1 \) \( v_{1i} \) \( m_2 \) \( v_{2i}=0 \) ★ COLLISION AFTER \( m_1 \) \( v_{1f} \) \( m_2 \) \( v_{2f} \) \( m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f} \)

The total momentum \( \vec{p}_{\text{total}} \) is the same before and after the collision. Momentum lost by one object is gained by the other.

Elastic vs Inelastic Collisions

Elastic Inelastic Perfectly Inelastic
Momentum Conserved Conserved Conserved
Kinetic Energy Conserved Not conserved Max loss
Objects after Bounce apart May deform Stick together
Example Billiard balls Car crash (bounce) Cars crumple together

Perfectly Inelastic: Stick Together

BEFORE \( m_1 \) \( m_2 \) at rest AFTER \( m_1+m_2 \) stuck \( v_f \) KE is lost to heat, sound, and deformation

Elastic Collisions: Two Equations

In a perfectly elastic collision, both momentum and kinetic energy are conserved, giving us two simultaneous equations:

How to Solve Collision Problems

1
Identify the system: Which objects are involved? Is it isolated (no net external force)?
2
Choose a direction as positive: Typically rightward is \( + \). Assign signs to all velocities.
3
Write the conservation equation: \( m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f} \)
4
Identify the collision type: If objects stick together, use \( (m_1 + m_2) v_f \). If elastic, add the KE equation.
5
Plug in known values and solve for the unknown velocity (or mass).
6
Check your answer: Does the sign make sense? Is momentum conserved numerically?

Example 1: Perfectly Inelastic

A \( 5\,\text{kg} \) cart moving at \( 4\,\text{m/s} \) collides with a stationary \( 3\,\text{kg} \) cart. They stick together. Find the final velocity.

5 kg 4 m/s 3 kg v = 0 8 kg stuck \( v_f = ? \) \( (5)(4)+(3)(0)=(5+3)v_f \;\Rightarrow\; 20=8v_f \;\Rightarrow\; v_f=2.5\,\text{m/s} \)

Example 2: Head-On Inelastic

A \( 2\,\text{kg} \) ball moving at \( +6\,\text{m/s} \) collides head-on with a \( 4\,\text{kg} \) ball moving at \( -3\,\text{m/s} \). They stick together. Find \( v_f \).

Notice: The negative sign on \( v_{2i} \) was critical. Forgetting direction is the most common error in momentum problems.

Example 3: Elastic Collision

A \( 3\,\text{kg} \) ball at \( +8\,\text{m/s} \) elastically hits a stationary \( 1\,\text{kg} \) ball. Find both final velocities.

Special Elastic Cases

Equal Masses
The moving object stops; the stationary object moves with the original velocity. Think: Newton's cradle.
Heavy Hits Light
Heavy object barely slows down. Light object flies off at nearly \( 2v_{1i} \). Think: bowling ball vs ping-pong ball.
Light Hits Heavy
Light object bounces back at nearly the same speed. Heavy object barely moves. Think: ball thrown at a wall.

Common Mistakes

✕ Forgetting Signs
Velocity is a vector. Objects moving left should have negative velocity if right is positive. Dropping the sign is the most common error.
✕ Confusing KE with Momentum
KE is not always conserved in collisions. Only momentum is always conserved. KE conservation is an additional condition for elastic collisions only.
✕ Using Elastic Formulas for Inelastic
The shortcut formulas \( v_{1f} = \frac{m_1-m_2}{m_1+m_2}v_{1i} \) only apply to elastic collisions. For perfectly inelastic, objects share one \( v_f \).

Summary: Collision Types

Property Elastic Inelastic Perfectly Inelastic
Momentum Conserved Conserved Conserved
Kinetic Energy Conserved Partially lost Maximum loss
Final Velocities Two unknowns, two equations Two unknowns, need extra info One unknown: \( v_f \)
Key Equation \( p \) + \( KE \) conserved \( p \) conserved only \( (m_1+m_2)v_f \)
Objects after Separate Separate (deformed) Stick together

Question 1 of 5

A \( 6\,\text{kg} \) object moving at \( 3\,\text{m/s} \) collides with a \( 2\,\text{kg} \) stationary object. They stick together. What is their final velocity?

Question 2 of 5

In which type of collision is kinetic energy conserved?

Question 3 of 5

A \( 4\,\text{kg} \) ball at \( +5\,\text{m/s} \) and a \( 4\,\text{kg} \) ball at \( -3\,\text{m/s} \) collide and stick together. What is \( v_f \)?

Question 4 of 5

In an elastic collision, a \( 2\,\text{kg} \) ball at \( 10\,\text{m/s} \) hits a stationary \( 2\,\text{kg} \) ball. What happens?

Question 5 of 5

A \( 1500\,\text{kg} \) car at \( 20\,\text{m/s} \) rear-ends a \( 1000\,\text{kg} \) car at \( 10\,\text{m/s} \) (same direction). They lock bumpers. What is \( v_f \)?