Momentum Questions- Problem Solving and Examples

What Are Momentum Questions?

Momentum questions are a staple of physics exams. They test your ability to apply the conservation of momentum principle to solve problems involving moving objects.

If you can't crack these problems, you're going to struggle in any physics course that covers mechanics. There's no way around it.

The Momentum Formula You Need to Know

Momentum (p) = mass (m) × velocity (v)

That's it. One equation. But the math gets tricky when you start combining objects, collisions, and explosions.

Units matter here. Momentum is measured in kg·m/s. Keep your units consistent or your answers will be garbage.

Conservation of Momentum: The Core Principle

In a closed system with no external forces, total momentum before an event equals total momentum after.

m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'

The primes (') denote values after the collision or interaction.

Types of Momentum Problems

Elastic Collisions

Objects bounce off each other. Kinetic energy is conserved along with momentum. These are cleaner problems because you get two equations to work with.

Inelastic Collisions

Objects stick together or deform. Only momentum is conserved. Kinetic energy gets converted to other forms. You lose information, but the math is simpler.

Explosions

Objects separate from a stationary starting point. The total momentum before is zero, so the total momentum after must also be zero. This gives you a powerful constraint for solving unknowns.

Problem-Solving Strategy

• Identify all objects in the system
• Determine the time instant you're analyzing (before/after)
• Write momentum expressions for each object before and after
• Apply conservation of momentum equation
• Solve for the unknown variable
• Check your work for unit consistency

Worked Example 1: Basic Two-Object Collision

Problem: A 2 kg ball moving at 3 m/s collides with a stationary 4 kg ball. After the collision, the 2 kg ball stops. What is the velocity of the 4 kg ball?

Step 1: Identify known values

m₁ = 2 kg, v₁ = 3 m/s, m₂ = 4 kg, v₂ = 0 m/s, v₁' = 0 m/s

Step 2: Apply conservation of momentum

m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'

(2)(3) + (4)(0) = (2)(0) + (4)(v₂')

6 = 4v₂'

Step 3: Solve

v₂' = 1.5 m/s

The 4 kg ball moves forward at 1.5 m/s. Simple.

Worked Example 2: Inelastic Collision

Problem: A 3 kg object moving at 4 m/s collides with a 2 kg object moving at 2 m/s in the same direction. They stick together. Find their common velocity.

Set up the equation:

m₁v₁ + m₂v₂ = (m₁ + m₂)v'

(3)(4) + (2)(2) = (3 + 2)v'

12 + 4 = 5v'

16 = 5v'

v' = 3.2 m/s

The combined mass moves at 3.2 m/s. Notice the velocity is between the two initial values, which makes physical sense.

Worked Example 3: Explosion Problem

Problem: A 10 kg object at rest explodes into two fragments. Fragment A has mass 3 kg and moves at 20 m/s. What is the velocity of Fragment B?

Key insight: Initial momentum is zero because the object isn't moving.

0 = mₐvₐ + mᵦvᵦ

0 = (3)(20) + (7)(vᵦ)

0 = 60 + 7vᵦ

7vᵦ = -60

vᵦ = -8.57 m/s

The negative sign tells you Fragment B moves in the opposite direction. That's how explosions work—conservation of momentum forces opposite motion.

Worked Example 4: 2D Momentum Problem

Problem: A 2 kg ball moving east at 5 m/s collides with a 3 kg ball moving north at 4 m/s. They stick together. Find the final velocity.

You must solve each direction separately.

East direction (x-axis):

m₁v₁ = (m₁ + m₂)vₓ

(2)(5) = (5)vₓ

vₓ = 2 m/s

North direction (y-axis):

m₂v₂ = (m₁ + m₂)vᵧ

(3)(4) = (5)vᵧ

vᵧ = 2.4 m/s

Magnitude: v = √(2² + 2.4²) = √(4 + 5.76) = √9.76 = 3.12 m/s

Direction: tan⁻¹(2.4/2) = 50.2° north of east

Momentum vs. Kinetic Energy: Know the Difference

Common Mistakes That Will Kill Your Grade

Forgetting to account for direction. Momentum is a vector. If objects move in opposite directions, one velocity must be negative. Mess this up and you'll get answers that are physically impossible.

Using the wrong mass. After collisions where objects stick together, use the combined mass. Don't keep treating them as separate objects.

Mixing up elastic and inelastic. In elastic collisions, you can use kinetic energy conservation as a second equation. In inelastic ones, you can't. Students mix these up constantly.

Calculation errors in 2D problems. Solve x and y components independently. Then combine them at the end using Pythagorean theorem.

Getting Started: Your Action Plan

Stop reading and start practicing. Here's what you do:

1. Master the basic formula until you can write it without thinking
2. Solve 5 simple collision problems with one unknown
3. Move to inelastic collision problems where objects combine
4. Tackle explosion problems to build intuition about momentum transfer
5. Attempt 2D problems only after you're solid on 1D

Work through problems daily. Momentum questions don't get easier if you avoid them. They only get worse.

Quick Reference Formulas

• Basic momentum: p = mv
• Conservation: Σp_before = Σp_after
• Inelastic (combined mass): m₁v₁ + m₂v₂ = (m₁ + m₂)v'
• Explosion: 0 = Σmᵢvᵢ' (sum of all fragments)