How to Solve Elastic Collisions

Steel balls suspended in a cradle produce almost completely elastic collisions.

Elastic collisions are those in which both momentum and kinetic energy are conserved. No collisions in the real world are truly elastic -- they lose energy as sound, heat or work (such as the deformation of metal in a vehicle collision). One of the closest measurable examples is the desk toy that consists of steel balls swinging into each other. That's why it swings so long before stopping. Equations pertaining to the conservation of momentum and of kinetic energy will help solve elastic-collision problems.

Instructions

1. Consider two objects that collide. The first has mass of M1 and the second has a mass of M2. Initially, the first has momentum P1 and velocity V1. The second has momentum P2 and velocity V2. After they collide, the first object has momentum P3 and velocity V3 while the second has momentum P4 and velocity V4. Mass remains the same.

2. Define momentum as mass*velocity. The initial momentum of the first object is therefore M1*V1 and of the second M2*V2. After collision, the momentum of the first is M1*V3 and of the second is M2*V4. Because momentum is conserved: M1*V1 + M2*V2 = M1*V3 + M2*V4.

3. Solve a one-dimensional elastic collision, meaning that two objects collide head on. A car of 1,000 kilograms traveling at 10 meters/second (about 22 mph) hits a pedestrian of 50 kilograms (110 pounds). The car transfers a tenth of its momentum to the pedestrian. What happens? The initial momentum of the pedestrian is 0 m/s * 100 kg = 0 kg*m/s. The car's is 1,000 kg * 10 m/s = 10,000 kg*m/s. After the collision, the car has momentum of 9/10 * 10,000 kg*m/s which is 9,000 kg*m/s. The pedestrian has absorbed 1,000 kg*m/s of momentum. For the car, 9,000 kg*m/s = 1,000 kg * V3. Solving for V3, we get 9 m/s which is about 20 mph. For the pedestrian, 1,000 kg*m/s = 50 kg * V4 and V4 is 20 m/s (about 45 mph). In this case, the car will throw the pedestrian at twice its initial velocity.

4. Consider the case of the steel-ball toy. Each steel ball has the same mass, and momentum is retained. From the definition of momentum, we can see that therefore velocity is also retained -- if we assume the collision is perfectly elastic. If we drop one ball, the momentum will be transferred through the middle balls to the last ball on the other end, which will then fly up with the same velocity. If we drop two balls, two balls on the other end will fly up with the same velocity. In practice, energy is lost to friction and sound so the balls will not clack forever.

Tips & Warnings

• In one dimension, it is also possible to have one object bounce back from the other, in which case you need to consider negative values of velocity.

• For two-dimensional problems, you need to measure the angle of deflection and the X and Y values of the initial and final velocity vectors. For three-dimensional problems, throw in the Z axis.

• The actual transfer of momentum of one object to another is extraordinarily difficult to calculate in elastic collisions. It is usually a given in such problems.

References

• Hyperphysics: Elastic and Inelastic Collisions
• Kirupa: Elastic Collisions
• The Physics Classroom: Momentum and Collisions