Momentum in Physics: Complete UK GCSE and A-Level Guide
Momentum is one of the most important concepts in physics, linking force, motion, and time. It is defined as the product of an object's mass and its velocity and is measured in kilogram metres per second (kg·m/s). Momentum is a vector quantity - it has both magnitude and direction - making it essential for analysing collisions, explosions, and changes in motion.
The Momentum Formula: p = mv
The formula for linear momentum is p = mv, where:
- p = momentum (kg·m/s or N·s)
- m = mass (kg)
- v = velocity (m/s)
Rearrangements: m = p/v (to find mass) and v = p/m (to find velocity). Always include direction when stating momentum - a car travelling at 20 m/s eastward has a different momentum to one moving westward at the same speed.
Worked Example: Momentum Calculation
A football of mass 0.45 kg is kicked and travels at 15 m/s. Calculate its momentum.
- p = m × v = 0.45 × 15 = 6.75 kg·m/s (in the direction of travel)
If the same ball is heading toward you at 15 m/s and you take it as the negative direction: p = 0.45 × (-15) = -6.75 kg·m/s.
Impulse: J = FΔt
Impulse is the product of force and the time for which it acts. The impulse-momentum theorem states:
J = F × Δt = Δp = m(v − u)
Impulse equals the change in momentum. This has important real-world applications:
- Car safety: Crumple zones increase collision time Δt, reducing peak force F for the same impulse (change in momentum).
- Cricket/baseball: Following through when hitting a ball increases contact time, increasing the impulse and therefore the change in momentum (making the ball go faster).
- Airbags: Increase the time over which the driver decelerates, reducing the force on their body.
- Foam packaging: Increases impact time when dropping fragile objects, reducing peak force.
Conservation of Momentum
The law of conservation of momentum states that the total momentum of a closed system remains constant if no external net force acts on it. For two objects colliding:
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
where u represents initial velocities and v represents final velocities.
Types of Collisions
Perfectly Inelastic Collision
The objects stick together after collision. Momentum is conserved but kinetic energy is not (some converts to thermal energy and sound). The formula becomes:
m₁u₁ + m₂u₂ = (m₁ + m₂)v
Example: A 3 kg trolley moving at 4 m/s collides with a stationary 1.5 kg trolley and they stick together. Final velocity: v = (3×4 + 1.5×0)/(3+1.5) = 12/4.5 = 2.67 m/s.
Elastic Collision
Both momentum and kinetic energy are conserved. The objects bounce apart. For equal masses in a 1D elastic collision, the velocities are exchanged (as seen in Newton's cradle). The general formulas for final velocities are:
- v₁ = [(m₁ - m₂)u₁ + 2m₂u₂] / (m₁ + m₂)
- v₂ = [(m₂ - m₁)u₂ + 2m₁u₁] / (m₁ + m₂)
Newton's Second Law and Momentum
Newton's second law is more accurately stated as: Force = rate of change of momentum, or F = Δp/Δt. This is the original form Newton used and it is more general than F = ma (which assumes constant mass). It is especially important for:
- Rocket propulsion (mass decreases as fuel burns)
- Variable mass systems
- Understanding why impulse equals change in momentum
Real-World Applications of Momentum
Momentum is fundamental to many areas of physics and engineering:
- Vehicle collisions: Police use momentum conservation to reconstruct accidents.
- Rocket science: Conservation of momentum explains rocket thrust (exhaust gases go backward, rocket goes forward).
- Nuclear physics: Particle collisions in accelerators follow conservation of momentum.
- Sport: Understanding momentum improves technique in cricket, football, tennis, and billiards.
- Space travel: Spacecraft use momentum transfer with gravitational slingshots to gain speed.
Momentum and Kinetic Energy Comparison
Students sometimes confuse momentum (p = mv) with kinetic energy (KE = ½mv²). Key differences:
- Momentum is always conserved in collisions; kinetic energy is only conserved in elastic collisions.
- Momentum is a vector; kinetic energy is a scalar.
- Units differ: momentum in kg·m/s, kinetic energy in joules (J).
- The relationship between them: KE = p²/(2m) or p = √(2m × KE).