Momentum is a measure of how hard it is to stop a moving object. It depends on both how much mass the object has and how fast it moves. In Physics you use momentum to analyze collisions in one dimension, and to evaluate devices, such as crumple zones, that reduce the force on people in a crash. The central idea is simple: in a system with no net outside force, the total momentum does not change.
Definitions
- Momentum: p = m · v, in kg·m/s. It is a vector, so direction matters. Choose one direction as positive and give the other a negative sign.
- Impulse: the force on an object multiplied by the time it acts, F · Δt. Impulse equals the change in momentum: F · Δt = Δp.
Why total momentum is conserved
When two objects collide, each pushes on the other with a force. By Newton's third law the forces are equal in size and opposite in direction, and they act for exactly the same time. So the impulses are equal and opposite: one object gains as much momentum as the other loses. The total of the two stays the same. This holds as long as no net outside force, such as friction, acts on the pair during the collision.
Worked example 1: carts that stick together
A 2.0 kg cart moves at +3.0 m/s and hits a 1.0 kg cart at rest. They stick together. What is their speed afterward?
- Momentum before: 2.0 kg × 3.0 m/s + 1.0 kg × 0 = 6.0 kg·m/s.
- After the collision the combined mass is 3.0 kg moving at v: 3.0 kg × v = 6.0 kg·m/s.
- v = 2.0 m/s, in the original direction.
Check the kinetic energy. Before: ½ × 2.0 × 3.0² = 9.0 J. After: ½ × 3.0 × 2.0² = 6.0 J. Momentum was conserved, but 3.0 J of kinetic energy became thermal energy and deformation. Collisions in which objects stick together are inelastic.
Worked example 2: pushing apart
Two skaters, 60 kg and 40 kg, stand at rest on ice. The total momentum is zero. The 60 kg skater pushes the other, who moves off at +3.0 m/s.
- Momentum of the lighter skater: 40 kg × 3.0 m/s = +120 kg·m/s.
- The total must remain zero, so the other skater has −120 kg·m/s: 60 kg × v = −120 kg·m/s.
- v = −2.0 m/s. The heavier skater moves at 2.0 m/s in the opposite direction.
The lighter skater ends up with the greater speed, but the two momenta cancel exactly.
Worked example 3: designing for a safer stop
A 1000 kg car traveling at 20 m/s stops completely, so its change in momentum has size 1000 kg × 20 m/s = 2.0 × 10⁴ kg·m/s. The average force depends on how long the stop takes.
- Stopping in 0.10 s (hitting a rigid wall): F = 2.0 × 10⁴ ÷ 0.10 = 2.0 × 10⁵ N.
- Stopping in 0.50 s (a crumple zone that collapses gradually): F = 2.0 × 10⁴ ÷ 0.50 = 4.0 × 10⁴ N.
The momentum change is the same, but spreading it over five times as long cuts the average force to one-fifth. This is the principle behind crumple zones, air bags and padded dashboards. A good design lengthens the time of the collision.
Using the idea as a checking tool
When you set up a collision problem, write down the momentum of every object before and after, with signs. Set the totals equal, then solve for the unknown. Afterward, check that the result is sensible: a stuck-together pair cannot move faster than the faster object started, and the final kinetic energy cannot exceed the initial kinetic energy.
Common mistakes
- Ignoring direction. Objects moving in opposite directions have momenta of opposite sign. Add them with signs.
- Confusing momentum with kinetic energy. Momentum is conserved in every collision of an isolated system. Kinetic energy is conserved only in elastic ones.
- Mixing up momentum and force. A moving object has momentum, but a force is needed only to change it.
- Forgetting units. Momentum is in kg·m/s, which is the same as N·s.
- Assuming a heavier object always wins. In a collision both objects feel equal-sized forces. The lighter one just changes velocity more.
Practice Physics questions with an explanation for every answer, or read the guide to California Physics practice.
Source: the California Next Generation Science Standards (CA NGSS), as set out in the California Science Framework, HS-PS2-2, HS-PS2-3.