Collisions & Momentum
Two objects collide head-on. Their total momentum stays the same; how much they bounce apart decides where the kinetic energy goes. Elastic, sticking together, or anything in between.
What is Momentum?
Momentum p = m·v is mass in motion, with a direction. In a collision the two objects push on each other equally hard for the same short time (Newton’s third law), so whatever momentum one gains, the other loses: the total stays the same.
Kinetic energy is a different story. In an elastic collision (billiard balls, almost) it is kept; in an inelastic one part turns into heat, sound and dents. When the objects stick together, the most energy that momentum allows is lost.
Collision, Step by Step
pKey Relations
And the same with 1 and 2 swapped for v2′. With e = 1 this is the elastic collision, with e = 0 both end at the common velocity.
The common velocity is also the velocity of the centre of mass, which no collision can change.
Only the motion relative to the centre of mass can be lost; the share that goes is 1 − e².
Sources & Verification
- D. Halliday, R. Resnick & J. Walker, Fundamentals of Physics, 10th ed., Wiley, 2014 — Ch. 9 (Center of Mass and Linear Momentum).
Frequently Asked Questions
How do you calculate the velocities after a collision?
Use two rules: the total momentum m₁v₁ + m₂v₂ stays the same, and the objects separate at e times the speed at which they closed in (e = 1 elastic, e = 0 they stick). For 2 kg at 3 m/s hitting 1 kg at −1 m/s elastically, that gives 0.333 m/s and 4.33 m/s afterwards.
What is the difference between an elastic and an inelastic collision?
In an elastic collision the kinetic energy is kept: the objects bounce apart as fast as they came together. In an inelastic collision part of it turns into heat, sound and deformation. Momentum is conserved in both. When the objects stick together (perfectly inelastic), the most energy is lost.
What is the coefficient of restitution?
e is the speed at which two objects fly apart divided by the speed at which they came together, from 0 (they stick) to 1 (perfectly elastic). For a ball dropped on a hard floor, e = √(bounce height ÷ drop height). A tennis ball dropped from 254 cm must bounce 135–147 cm, so e is about 0.73 to 0.76.
Is momentum always conserved in a collision?
Yes, as long as no outside force acts. During the short moment of a collision the forces between the two objects are so large that friction or gravity hardly matter, so the total momentum just before equals the total just after. Kinetic energy is only conserved in an elastic collision.
How fast does a block move when a bullet hits it?
The bullet sticks, so v = m₁v₁ ÷ (m₁ + m₂). A 10 g bullet at 400 m/s into a 2 kg wooden block gives 4 ÷ 2.01 = 1.99 m/s. Of the bullet's 800 J of kinetic energy only 4 J is left: 99.5% goes into heat and splinters. This is the principle of the ballistic pendulum.