
Imagine a collision between two objects with masses m_{1} and m_{2}. The two objects have initial velocities v_{i1} and v_{i2} and final velocities v_{f1} and v_{f2}. Momentum conservation for these two colliding objects can be written as

(11.3) 
$${m}_{1}{v}_{i1}+{m}_{2}{v}_{i2}={m}_{1}{v}_{f1}+{m}_{2}{v}_{f2}$$
 m_{1}  =  mass of object 1 (kg)  v_{i1}  =  initial velocity of object 1 (m/s)  v_{f1}  =  final velocity of object 1 (m/s)  m_{2}  =  mass of object 2 (kg)  v_{i2}  =  initial velocity of object 2 (m/s)  v_{f2}  =  final velocity of object 2 (m/s) 
 Conservation of momentum (two objects)


There are two basic types of collisions in physics: elastic and inelastic. In an inelastic collision, some of the initial kinetic energy of the objects is transformed into heat and/or works to deform the shape of the objects. Auto collisions are nearly always inelastic, because of the damage caused to the cars. In the special case of a perfectly inelastic collision, the two objects stick together after impact.

Remember the example on page 622 of an astronaut who threw a wrench, causing her to move away in the opposite direction? That problem is basically a perfectly inelastic collision played in reverse! Imagine taking a video of the astronaut throwing the wrench, but then playing the video backward. The astronaut and the wrench will be moving toward each other until the astronaut catches the wrench and they both stop in place. The physics of momentum conservation underlies both possible events. Many physical phenomena (but not all) exhibit this kind of symmetry or reversibility.


A perfectly inelastic collision is depicted in the illustration above. These collision problems are solved in the same way as any other collision problem, using the conservation of momentum. Moreover, in the perfectly inelastic collision case the final velocities of the two objects are set to be equal—because the objects stick together!

A 100 kg hockey player, moving at 2 m/s, collides headon with a 75 kg hockey player moving at 1 m/s. After impact, they become entangled and slide together. What is their velocity after impact?
Asked: 
final velocity v_{f} after collision 
Given: 
m_{1} = 100 kg, m_{2} = 75 kg, v_{i1} = +2 m/s, v_{i2} = −1 m/s, v_{f1} = v_{f2} = v_{f}, since they move together after the (inelastic) collision 
Relationships: 
Conservation of momentum: m_{1}v_{i1} + m_{2}v_{i2} = m_{1}v_{f1} + m_{2}v_{f2} 
Solution: 
Equate the final velocities: m_{1}v_{i1} + m_{2}v_{i2} = (m_{1} + m_{2})v_{f}. Divide both sides by (m_{1} + m_{2}): $$\frac{{m}_{1}{v}_{i1}+{m}_{2}{v}_{i2}}{{m}_{1}+{m}_{2}}=\frac{\left({m}_{1}+{m}_{2}\right){v}_{f}}{{m}_{1}+{m}_{2}}$$ Cancel terms to solve for v_{f} and substitute the values: $${v}_{f}=\frac{{m}_{1}{v}_{i1}+{m}_{2}{v}_{i2}}{{m}_{1}+{m}_{2}}=\frac{\left(100\text{kg}\right)\left(2\text{m/s}\right)+\left(75\text{kg}\right)\left(1\text{m/s}\right)}{100\text{kg}+75\text{kg}}=+0.71\text{m/s}$$ 
Answer: 
v_{f} = +0.71 m/s, in the same direction as the 100 kg player. 

Two cars collide headon, partially crumpling the front end of each. The cars bounce off each other from the collision, ending 1.3 m apart. What type of collision is this?
 elastic
 inelastic
 perfectly inelastic
 There is not enough information to determine the type of collision.

The correct answer is b, inelastic. Some of the kinetic energy of the cars went into crumpling their front ends; therefore, this is an inelastic collision. The cars were not stuck together after impact, so it is not a perfectly inelastic collision.

Two 1 kg balls, traveling at +1 m/s and −1 m/s, respectively, collide with each other and stick together after impact. What is their velocity after the collision?
 −1 m/s
 0 m/s
 +1 m/s
 +2 m/s

The correct answer is b, 0 m/s. The first ball has a momentum before the collision of p_{1} = mv_{1} = (1 kg)×(−1 m/s) = −1 kg m/s, while the second ball has a momentum p_{2} = (1 kg)×(+1 m/s) = +1 kg m/s. The total momentum before the collision is therefore zero! From momentum conservation, the total momentum after the collision—with the balls stuck together—is also zero. Their velocity must therefore be zero.
