Apply the principle of the conservation of momentum to solve simple problems in one dimension

Momentum – IGCSE Physics 0625 (Topic 1.6)

Objective

Apply the principle of conservation of momentum to solve simple one‑dimensional problems and understand its broader context (vector nature, Newton’s 2nd law in momentum form, extensions and practical applications).

Key Definitions

• Momentum ( p ) – a vector quantity defined by

  p = m v

  where m is mass (kg) and v is velocity (m s⁻¹).

  Direction must be indicated with a sign (±) or an arrow.

  SI unit: kilogram‑metre per second (kg m s⁻¹).

• Impulse ( J ) – the change in momentum produced by a force acting for a time interval:

  J = Δp = F Δt.

  Impulse has the same unit as momentum (kg m s⁻¹ or N·s). Useful for collisions that occur over a short time.

• Newton’s 2nd law in momentum form –

  F = Δp / Δt (or F Δt = Δp).

  This links the net external force on a system to the rate of change of its momentum.

Conservation of Momentum (One Dimension)

In an isolated system (no external forces), the total momentum before an interaction equals the total momentum after the interaction.

For two objects of masses m₁ and m₂ moving along a straight line:

Choose a sign convention (e.g., rightward = + , leftward = –) and treat momentum as a signed quantity.

Types of Collisions

Step‑by‑Step Method for Solving One‑Dimensional Momentum Problems

1. Read & sketch: draw a clear diagram showing all objects, their masses, and the direction of motion.
2. List known quantities: write each mass and velocity (include sign for direction).
3. Choose a sign convention: e.g., rightward = + , leftward = – .
4. Write the conservation equation:

   Σ m v (initial) = Σ m v (final)

5. If the collision is elastic, also write the kinetic‑energy equation:

   ½ m₁v₁ᵢ² + ½ m₂v₂ᵢ² = ½ m₁v₁𝒻² + ½ m₂v₂𝒻²

6. Solve the simultaneous equations for the unknown velocity (or velocities).
7. Check: verify sign, magnitude, and physical plausibility (e.g., total kinetic energy should not increase in an inelastic collision).

Common Pitfalls

• Forgetting that momentum is a vector – always keep track of direction with signs or arrows.
• Mixing up initial and final velocities in the conservation equation.
• Assuming kinetic energy is conserved in an inelastic collision.
• Omitting any object that belongs to the isolated system (the Earth’s reaction force is negligible and can be ignored).
• Changing the sign convention part‑way through a problem.

Worked Example – Perfectly Inelastic Collision

Problem: A 0.5 kg cart moving right at 4 m s⁻¹ collides head‑on with a 0.8 kg cart moving left at 2 m s⁻¹. The carts stick together. Find their common velocity after the collision.

1. Choose rightward as positive.

   • v₁ᵢ = +4 m s⁻¹ (0.5 kg cart)
   • v₂ᵢ = –2 m s⁻¹ (0.8 kg cart)

2. Total mass after sticking: mₜ = 0.5 kg + 0.8 kg = 1.3 kg.
3. Apply conservation of momentum:

   (0.5)(+4) + (0.8)(–2) = (1.3) v𝒻

   2.0 – 1.6 = 1.3 v𝒻

   0.4 = 1.3 v𝒻

   v𝒻 = 0.4 / 1.3 ≈ 0.31 m s⁻¹

4. Result: The combined carts move to the right at ≈ 0.31 m s⁻¹.

Suggested Diagram

Extension – Beyond the Core Syllabus

• Resultant of two perpendicular vectors (useful for the supplementary syllabus point on vector addition).

  
  If p₁ = m₁v₁ is along the x‑axis and p₂ = m₂v₂ is along the y‑axis, the magnitude of the total momentum is

  
  p = √(p₁² + p₂²) and its direction is given by tan θ = p₂ / p₁.

• Momentum of a system with more than two objects.

  
  For n bodies:

  
  Σ mᵢvᵢᵢ = Σ mᵢvᵢ𝒻 (i = 1 … n).

  Example: three carts colliding sequentially – treat the three‑cart system as a single isolated system.

• Real‑world applications (connect theory to everyday life).

  • Car‑crash safety: crumple zones increase the time Δt, reducing the force F for a given change in momentum.
  • Sports: a boxer’s punch transfers momentum to the bag; the impulse felt is related to the change in the bag’s momentum.
  • Rocket propulsion: gases expelled backward give the rocket forward momentum (conservation of momentum of the rocket‑gas system).

Practical Activity – Measuring Momentum in a One‑Dimensional Collision

Goal: Verify conservation of momentum using two colliding carts on a low‑friction track.

Method (outline)

1. Set up the track horizontally; ensure carts can move freely.
2. Measure and record the mass of each cart (to 0.01 kg).
3. Place Cart A at the left end, Cart B at the right end. Use the motion sensor to record the speed of each cart just before impact (initial velocities).
4. Allow the carts to collide (choose elastic or perfectly inelastic by attaching a Velcro strip for sticking).
5. Immediately after the collision, record the final speed(s) of the cart(s).
6. Repeat the experiment three times for each collision type and calculate average values.

Data‑table template

Guiding questions (AO3)

• Do the measured total momenta before and after the collision agree within experimental uncertainty? Explain any discrepancy.
• How does increasing the collision time (e.g., using a soft bumper) affect the force experienced by the carts? Relate this to F = Δp/Δt.
• Compare the kinetic‑energy change for elastic and perfectly inelastic collisions. Why is kinetic energy not conserved in the latter?

Practice Questions

1. A 1.2 kg ball moving at 3 m s⁻¹ collides elastically with a stationary 0.8 kg ball. Find the velocities of both balls after the collision.
2. A 2.0 kg sled moving at 5 m s⁻¹ picks up a 0.5 kg snowball that was initially at rest. What is the speed of the sled‑snowball system after the collision?
3. Two ice skaters, A (mass 50 kg) and B (mass 70 kg), push off each other. If A moves away at 2 m s⁻¹, what is B’s speed?
4. A 0.3 kg ball strikes a 0.7 kg ball moving opposite direction at 1 m s⁻¹. The collision is perfectly elastic. Determine the final velocities of both balls.
5. A 1.5 kg cart moving at 2 m s⁻¹ collides with a 2.0 kg cart moving at 1 m s⁻¹ in the same direction. After the collision they stick together. Find the common speed.

Quick Reference Sheet

Syllabus Alignment – Quick Review