GCSE — physicsHard6 min read

Momentum and Conservation

p = mv; conservation of momentum; collisions and explosions

Tutor AI Team
Tutor AI Team

# Momentum and Conservation — GCSE Physics

Momentum is a fundamental quantity in physics that combines mass and velocity. The law of conservation of momentum is one of the most powerful principles in physics — it allows us to predict the outcomes of collisions and explosions.

1. What Is Momentum?

p=m×v\boxed{p = m \times v}

Where:

  • pp = momentum (in kg m/s)
  • mm = mass (in kg)
  • vv = velocity (in m/s)

Momentum is a vector quantity — it has both magnitude and direction.

Key Points

  • A stationary object has zero momentum (v=0v = 0)
  • A heavier object moving at the same speed has more momentum
  • A faster object of the same mass has more momentum
  • Momentum can be positive or negative (depending on direction)

2. Conservation of Momentum

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

Total momentum before=Total momentum after\text{Total momentum before} = \text{Total momentum after}

m1u1+m2u2=m1v1+m2v2m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2

This applies to:

  • Collisions (objects hitting each other)
  • Explosions (objects moving apart)
  • Any interaction in a closed system

Why Is Momentum Conserved?

Newton's Third Law: when two objects interact, they exert equal and opposite forces for the same time. Since impulse (F×tF \times t) equals change in momentum, the momentum gained by one object equals the momentum lost by the other.

3. Types of Collisions

Elastic Collisions

  • Total momentum is conserved ✓
  • Total kinetic energy is conserved ✓
  • Objects bounce apart
  • Example: billiard balls (approximately elastic)

Inelastic Collisions

  • Total momentum is conserved ✓
  • Total kinetic energy is NOT conserved ✗ (some converted to heat, sound, deformation)
  • Most real collisions are inelastic
  • In a perfectly inelastic collision, the objects stick together and move as one

4. Explosions

In an explosion, objects that were initially together (and stationary) move apart.

Before: Total momentum = 0 (everything is stationary)

After: Total momentum must still = 0

So: m1v1+m2v2=0m_1 v_1 + m_2 v_2 = 0, which means m1v1=−m2v2m_1 v_1 = -m_2 v_2

The objects move in opposite directions with momenta that are equal in magnitude.

Worked Example: Calculating Momentum

Problem

Question: Calculate the momentum of a 1200 kg car moving at 15 m/s.

Solution

p=mv=1200×15=18,000 kg m/sp = mv = 1200 \times 15 = 18{,}000 \text{ kg m/s}

Worked Example: Collision (Sticking Together

Problem

Question: A 2 kg ball moving at 6 m/s hits a stationary 4 kg ball. They stick together. Calculate their velocity after the collision.

Solution

m1u1+m2u2=(m1+m2)vm_1 u_1 + m_2 u_2 = (m_1 + m_2) v 2×6+4×0=(2+4)×v2 \times 6 + 4 \times 0 = (2 + 4) \times v 12=6v12 = 6v v=2 m/sv = 2 \text{ m/s}

Worked Example: Collision (Bouncing Apart

Problem

Question: A 3 kg trolley A moves at 4 m/s and hits a stationary 1 kg trolley B. After the collision, trolley A moves at 1 m/s in the same direction. Find the velocity of trolley B.

Solution

mAuA+mBuB=mAvA+mBvBm_A u_A + m_B u_B = m_A v_A + m_B v_B 3×4+1×0=3×1+1×vB3 \times 4 + 1 \times 0 = 3 \times 1 + 1 \times v_B 12=3+vB12 = 3 + v_B vB=9 m/sv_B = 9 \text{ m/s}

Worked Example: Explosion

Problem

Question: A 5 kg cannon fires a 0.5 kg cannonball at 100 m/s. Calculate the recoil velocity of the cannon.

Solution

Before: total momentum = 0 (both stationary) 0=mcannonvcannon+mballvball0 = m_{\text{cannon}} v_{\text{cannon}} + m_{\text{ball}} v_{\text{ball}} 0=5×vcannon+0.5×1000 = 5 \times v_{\text{cannon}} + 0.5 \times 100 0=5vcannon+500 = 5v_{\text{cannon}} + 50 vcannon=−10 m/sv_{\text{cannon}} = -10 \text{ m/s}

The cannon recoils at 10 m/s in the opposite direction to the cannonball.

Worked Example: Determining Collision Type

Problem

Question: A 2 kg ball at 5 m/s hits a 2 kg ball at rest. After the collision, the first ball stops and the second moves at 5 m/s. Show that momentum is conserved. Is this elastic?

Solution

Before: p=2×5+2×0=10p = 2 \times 5 + 2 \times 0 = 10 kg m/s After: p=2×0+2×5=10p = 2 \times 0 + 2 \times 5 = 10 kg m/s ✓ Conserved

KE before: 12(2)(52)=25\frac{1}{2}(2)(5^2) = 25 J KE after: 12(2)(52)=25\frac{1}{2}(2)(5^2) = 25 J ✓ KE conserved

This is an elastic collision.

6. Momentum and Safety

Momentum principles explain many safety features:

Crumple Zones

Car crumple zones increase the time over which the car decelerates. Since F=Δp/ΔtF = \Delta p / \Delta t, increasing time reduces the force on passengers.

Air Bags

Air bags also increase the time for the passenger's momentum to decrease, reducing the force on the body.

Seatbelts

Stretchy seatbelts extend the time of deceleration, reducing the force.

Key relationship: F=ΔpΔt=mv−mutF = \frac{\Delta p}{\Delta t} = \frac{mv - mu}{t}

Small force = long time. Large force = short time. (For the same change in momentum.)

7. Practice Questions

    1. Calculate the momentum of a 0.15 kg cricket ball moving at 40 m/s. (2 marks)
    1. A 5 kg trolley moving at 3 m/s collides with and sticks to a stationary 10 kg trolley. Calculate the velocity after the collision. (3 marks)
    1. A bullet of mass 0.01 kg is fired at 400 m/s into a 2 kg block of wood on a smooth surface. The bullet embeds in the block. Calculate the velocity of the block and bullet together. (3 marks)
    1. Two ice skaters push apart from rest. Skater A (60 kg) moves left at 2 m/s. Calculate the velocity of Skater B (40 kg). (3 marks)
    1. Explain how crumple zones reduce injury in a car crash. Use the concept of momentum in your answer. (4 marks)

    Answers

Want to check your answers and get step-by-step solutions?

Get it on Google PlayDownload on the App Store

Frequently Asked Questions

Is momentum always conserved?

Momentum is conserved in a closed system (no external forces). If external forces act (like friction from the floor), the system is not closed and total momentum may change.

What's the difference between momentum and kinetic energy?

Momentum (p=mvp = mv) is a vector — it has direction. Kinetic energy (KE=12mv2KE = \frac{1}{2}mv^2) is a scalar — it has no direction. Momentum is always conserved in collisions; kinetic energy is only conserved in elastic collisions.

Can momentum be negative?

Yes. If you define rightwards as positive, then an object moving leftwards has negative momentum. This is essential for solving collision problems.

Summary

  • Momentum = mass × velocity: p=mvp = mv (kg m/s, vector)
  • Conservation of momentum: total momentum before = total momentum after (in closed systems)
  • Collisions: elastic (KE conserved) or inelastic (KE not conserved)
  • Explosions: total initial momentum = 0, objects move in opposite directions
  • Safety: crumple zones, airbags, seatbelts increase time → reduce force (F=Δp/ΔtF = \Delta p / \Delta t)
Tags:
#gcse#physics#forces#momentum#conservation#collisions

Related Topics

Ready to Ace Your GCSE physics?

Get instant step-by-step solutions to any problem. Snap a photo and learn with Tutor AI — your personal exam prep companion.

Get it on Google PlayDownload on the App Store