Inelastic Collisions in One Dimension

Read this text. As we learned in the previous video, in reality, no collisions are perfectly elastic because some kinetic energy is always "lost" by being converted to other forms of energy. Another example of an elastic collision is if two balls collide on a smooth icy surface. Because the ice has almost no friction, little kinetic energy would be lost to friction.

See an example of two blocks experiencing a totally inelastic collision in Figure 8.8. See a good example of an inelastic collision in Figure 8.9. In this example, a hockey goalie stops a puck in the net. Although the ice surface is essentially frictionless, some kinetic energy of the puck is converted to heat and sound as the goalie stops it. A totally inelastic collision (also called a perfectly inelastic collision) is an inelastic collision where the objects "stick together" upon colliding.

We have seen that in an elastic collision, internal kinetic energy is conserved. An inelastic collision is one in which the internal kinetic energy changes (it is not conserved). This lack of conservation means that the forces between colliding objects may remove or add internal kinetic energy. Work done by internal forces may change the forms of energy within a system. For inelastic collisions, such as when colliding objects stick together, this internal work may transform some internal kinetic energy into heat transfer. Or it may convert stored energy into internal kinetic energy, such as when exploding bolts separate a satellite from its launch vehicle.

Inelastic Collision

An inelastic collision is one in which the internal kinetic energy changes (it is not conserved).

Figure 8.8 shows an example of an inelastic collision. Two objects that have equal masses head toward one another at equal speeds and then stick together. Their total internal kinetic energy is initially \frac{1}{2} m v^{2}+\frac{1}{2}
    m v^{2}=m v^{2} . The two objects come to rest after sticking together, conserving momentum. But the internal kinetic energy is zero after the collision. A collision in which the objects stick together is sometimes called a perfectly inelastic collision because it reduces internal kinetic energy more than does any other type of inelastic collision. In fact, such a collision reduces internal kinetic energy to the minimum it can have while still conserving momentum.

Perfectly Inelastic Collision

A collision in which the objects stick together is sometimes called "perfectly inelastic".

The system of interest contains two equal masses with mass m. One moves to the right and the other moves to the left with the same magnitude of velocity represented by V. Due to this their total momentum and net force remains zero. The internal kinetic energy is mv power 2. After collision the system of interest has no net velocity, no total momentum and no internal kinetic energy. This is true for all inelastic collisions.

Figure 8.8 An inelastic one-dimensional two-object collision. Momentum is conserved, but internal kinetic energy is not conserved. (a) Two objects of equal mass initially head directly toward one another at the same speed. (b) The objects stick together (a perfectly inelastic collision), and so their final velocity is zero. The internal kinetic energy of the system changes in any inelastic collision and is reduced to zero in this example.

Take-Home Experiment—Bouncing of Tennis Ball

  1. Find a racquet (a tennis, badminton, or other racquet will do). Place the racquet on the floor and stand on the handle. Drop a tennis ball on the strings from a measured height. Measure how high the ball bounces. Now ask a friend to hold the racquet firmly by the handle and drop a tennis ball from the same measured height above the racquet. Measure how high the ball bounces and observe what happens to your friend's hand during the collision. Explain your observations and measurements.

  2. The coefficient of restitution (c) is a measure of the elasticity of a collision between a ball and an object, and is defined as the ratio of the speeds after and before the collision. A perfectly elastic collision has a c of 1. For a ball bouncing off the floor (or a racquet on the floor), c can be shown to be c=(h / H)^{1 / 2} where h is the height to which the ball bounces and H is the height from which the ball is dropped. Determine c for the cases in Part 1 and for the case of a tennis ball bouncing off a concrete or wooden floor (c=0.85 for new tennis balls used on a tennis court).


Source: Rice University,
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Last modified: Friday, October 22, 2021, 1:49 PM