Momentum
Momentum
Definition: A measure of an object’s motion, equal to the product of its mass and velocity, that is conserved in isolated systems.
How It Works
- Momentum, p = mv, is a vector quantity, pointing in the same direction as the object’s velocity, with a magnitude that scales linearly with both mass and speed.
- Newton’s second law is more fundamentally stated as F = dp/dt, force equals the rate of change of momentum; F = ma is just the special case where mass stays constant.
- Impulse, J = FΔt, equals the change in momentum an object experiences, the impulse-momentum theorem, linking how hard and how long a force acts to the resulting change in motion.
- In any isolated system, with no net external force, total momentum before an interaction (a collision, explosion, or separation) equals total momentum after.
- Momentum conservation holds even when kinetic energy is not conserved, a crucial distinction from energy: momentum survives inelastic collisions where kinetic energy is lost to heat and deformation.
- Elastic collisions conserve both momentum and kinetic energy. Inelastic collisions conserve momentum but not kinetic energy. Perfectly inelastic collisions, where objects stick together, lose the maximum possible kinetic energy while still conserving momentum.
- Rocket propulsion works entirely on momentum conservation: expelling mass (exhaust) backward at high speed gives the rocket an equal and opposite forward momentum change.
- Angular momentum, L = Iω, is the rotational analog of linear momentum, conserved in the absence of external torque, which is why a figure skater spins faster by pulling their arms in.
- Total momentum of a multi-object system equals its total mass times the velocity of its center of mass; internal forces between objects (collision forces, explosion forces) can never change the system’s total momentum, only external forces can.
- Momentum conservation is a direct consequence of a deep symmetry of nature, the fact that the laws of physics look the same no matter where in space an experiment is performed (spatial translation symmetry).
Types of Collisions
| Type | Momentum conserved? | Kinetic energy conserved? | Example |
|---|---|---|---|
| Elastic | Yes | Yes | Billiard balls, gas molecule collisions |
| Inelastic | Yes | No, some converts to heat/sound/deformation | Two cars crumpling on impact |
| Perfectly inelastic | Yes | No, maximum KE loss; objects move together afterward | A dart sticking into a target |
Illustration
Under the Hood
Key equations:
p = mvF = dp/dt (Newton's second law, general form)J = FΔt = Δp (impulse-momentum theorem)Σp_before = Σp_after (conservation of momentum, isolated system)m₁v₁ᵢ + m₂v₂ᵢ = m₁v₁f + m₂v₂f (two-body collision)Worked Example 1: Momentum of a moving car Given: mass m = 1000 kg, speed v = 20 m/s Step 1: p = mv = 1000 × 20 Answer: p = 20,000 kg·m/s
Worked Example 2: Impulse accelerating a stationary object Given: force F = 500 N applied for Δt = 0.2 s to an object at rest, m = 10 kg Step 1: Impulse: J = FΔt = 500 × 0.2 = 100 kg·m/s Step 2: Since Δp = J and the object started at rest: v_f = Δp/m = 100/10 Answer: v_f = 10 m/s
Worked Example 3: Perfectly inelastic collision Given: car A, m₁ = 1500 kg at v₁ᵢ = 15 m/s, collides with stationary car B, m₂ = 1200 kg, and they stick together Step 1: Conservation of momentum: m₁v₁ᵢ + m₂v₂ᵢ = (m₁ + m₂)v_f Step 2: (1500 × 15) + (1200 × 0) = 22,500 = 2700 × v_f Answer: v_f = 22,500/2700 ≈ 8.33 m/s, the shared velocity of the tangled wreckage immediately after impact.
Worked Example 4: Elastic collision between equal masses Given: billiard ball A, m = 0.17 kg, moving at v = 3 m/s, strikes stationary ball B of equal mass, elastic collision Step 1: For an elastic collision between equal masses, the incoming ball transfers all its velocity to the target ball Answer: Ball A stops completely (v_A,f = 0), and ball B moves off at v_B,f = 3 m/s, a special case that’s easy to verify on a real pool table.
Worked Example 5: Rocket recoil from expelled exhaust Given: rocket at rest, total mass 5000 kg, expels m_exhaust = 50 kg of gas backward at v_exhaust = 2000 m/s Step 1: Total momentum stays zero (started at rest): 0 = m_rocket·v_rocket − m_exhaust·v_exhaust Step 2: 0 = (4950)(v_rocket) − (50)(2000) Step 3: v_rocket = 100,000/4950 Answer: v_rocket ≈ 20.2 m/s, the rocket’s forward speed gained purely from conservation of momentum after a single burst of exhaust.
Worked Example 6: Angular momentum conservation in a spin Given: figure skater, initial moment of inertia I₁ = 4 kg·m², initial angular velocity ω₁ = 2 rad/s, pulls arms in to I₂ = 1 kg·m² Step 1: Conservation of angular momentum: I₁ω₁ = I₂ω₂ Step 2: ω₂ = I₁ω₁/I₂ = (4 × 2)/1 Answer: ω₂ = 8 rad/s, spinning four times faster after pulling the arms in, with no external torque required.
Why It Matters
- Vehicle crash analysis and safety engineering use momentum conservation to reconstruct collisions and design crumple zones that extend impact time, reducing peak force.
- Rocket and spacecraft propulsion design is fundamentally an application of momentum conservation, expelling propellant to generate thrust.
- Particle physicists at facilities like the LHC use momentum conservation to infer the existence and properties of particles too short-lived to detect directly.
- Wind turbine and propeller blade design accounts for the momentum transferred between the moving fluid (air or water) and the rotating blades.
- Sports like billiards, bowling, and martial arts rely on precise transfer of momentum between objects or bodies to achieve controlled outcomes.
- Ballistics experts use momentum and impulse calculations to analyze projectile behavior on impact.
- Ion thrusters and other advanced spacecraft propulsion systems maximize exhaust velocity to gain more momentum change per unit of propellant mass, extending mission range.
Common Pitfalls
- Confusing momentum (a vector, p = mv) with kinetic energy (a scalar, KE = ½mv²); they follow different conservation rules and aren’t interchangeable.
- Forgetting momentum is a vector quantity and adding speeds directly instead of accounting for direction, objects moving in opposite directions need opposite signs.
- Assuming kinetic energy is always conserved in a collision; that’s only true for perfectly elastic collisions, and most real-world collisions, like car crashes, are inelastic.
- Applying conservation of momentum to a single object experiencing an external force; conservation only holds for the total momentum of an isolated system with no net external force.
- Confusing impulse (a change in momentum, force times time) with force itself; the same impulse delivered over a longer time means a smaller peak force, the whole principle behind airbags and crumple zones.
- Forgetting that changing angular momentum requires torque, not just any force; a force applied directly through an object’s center of mass or rotation axis produces no torque and doesn’t change its spin.
Comparison
| Property | Momentum | Kinetic energy |
|---|---|---|
| Type | Vector | Scalar |
| Formula | p = mv | KE = ½mv² |
| Conserved in all collisions (isolated system)? | Yes | Only in elastic collisions |
| Scales with velocity | Linearly | With the square |
| Rotational analog | Angular momentum, L = Iω | Rotational KE, ½Iω² |
FAQ
Is momentum always conserved? Only for an isolated system with no net external force; the moment an outside force acts (like friction from the ground), total momentum can change.
Can momentum be conserved when kinetic energy isn’t? Yes, and this is exactly what happens in every inelastic collision; momentum conservation is a stricter, more universal law than kinetic energy conservation in collisions.
Does a heavier or a faster object have more momentum? Neither automatically; momentum scales linearly with both mass and velocity, so a light, fast object can have the same momentum as a heavy, slow one.
Why does a gun recoil when fired? Conservation of momentum requires the bullet’s forward momentum to be matched by an equal and opposite momentum given to the gun; since the gun is far more massive than the bullet, it recoils at a much lower speed for the same magnitude of momentum.
How does momentum change at relativistic speeds? At speeds approaching the speed of light, momentum requires a relativistic correction, p = γmv, where the Lorentz factor γ grows without bound as v approaches c, meaning momentum diverges even though speed can’t exceed c.
Why do airbags and crumple zones reduce injury if the impulse is the same either way? They don’t change the total impulse (the change in momentum from crash speed to zero), but they extend the time over which it’s delivered; since impulse equals force times time, stretching the time out reduces the peak force experienced by the occupant.
Example
A Newton’s cradle, the classic desk toy with a row of suspended metal balls, demonstrates momentum and near-elastic energy conservation together: lifting and releasing one ball on one end transfers momentum, ball to ball, through the stationary middle balls, launching an equal number of balls off the far end at matching speed.
Related Terms
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