Elastic and Perfectly Inelastic Collisions Simulation


Q1) A fast-moving car hitting a haystack or hitting a cement wall produces vastly different results. 1. Do both experience the same change in momentum? 2. Do both experience the same impulse? 3. Do both experience the same force?
A. yes for all three
B. yes for 1 and 2
C. no for all three
D. no for 1 and 2

Answer) B.
Although stopping the momentum is the same whether done slowly or quickly, the force is vastly different. Be sure to distinguish between momentum, impulse, and force.

Q2) Freight car A is moving toward identical freight car B that is at rest. When they collide, both freight cars couple together. Compared with the initial speed of freight car A, the speed of the coupled freight cars is
A. the same.
B. half.
C. twice.
D. none of the above

Answer) B.
After the collision, the mass of the moving freight cars has doubled. Can you see that their speed is half the initial velocity of freight car A?

Elastic and Perfectly Inelastic Collisions Simulation

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Conservation of Momentum

The collision of objects clearly shows the conservation of momentum. Whenever objects collide in the absence of external forces, the net momentum of both objects before the collision equals the net momentum of both objects after the collision.

$\ net~ momentum _{before ~collision} = net~ momentum _{after ~collision} $

Elastic Collisions 
When a moving billiard ball collides head-on with a ball at rest, the first ball comes to rest and the second ball moves away with a velocity equal to the initial velocity of the first ball. We see that momentum is transferred from the first ball to the second ball. When objects collide without being permanently deformed and without generating heat, the collision is said to be an elastic collision. Colliding objects bounce perfectly in perfect elastic collisions, as shown in Figure 1. Note that the sum of the momentum vectors is the same before and after each collision.

FIGURE 1. Colliding objects bounce perfectly in elastic collisions. a. A moving ball strikes a ball at rest. b. Two moving balls collide head-on. c. Two balls moving in the same direction collide.

The total momentum is always constant throughout the collision. In addition, if the collision is perfectly elastic, the value of the total kinetic energy after the collision is equal to the value before the collision.

Perfectly Inelastic Collisions
A collision in which the colliding objects become distorted and generate heat during the collision is an inelastic collision. Momentum conservation holds true even in inelastic collisions. Whenever colliding objects become tangled or couple together, a totally inelastic collision occurs. Perfectly inelastic collision is a collision in which two objects
stick together after colliding. The freight train cars in Figure 2 provide an example. Suppose the freight cars are of equal mass m, and that one car moves at $\ 4 m/s $ toward the other car that is at rest. Can you predict the velocity of the coupled cars after impact? From the conservation of momentum, 

$\ net~ momentum _{before ~collision} = net~ momentum _{after ~collision} $

or, in equation form,

$\ (net~ mv)_{before} = (net~ mv)_{after} $
$\ (m)(4 ~m/s)~+~(m)(0~ m/s)~=~(2m)(v_{after}) $

Since twice as much mass is moving after the collision, can you see that the velocity, $\ v_{after} $, must be one half of $\ 4 m/s $? Solving for the velocity after the collision, we find $\ v_{after} = 2 m/s $ in the same direction as the velocity before the collision, $\ v_{before} $. The initial momentum is shared by both cars without loss or gain. Momentum is conserved.

FIGURE 2. In an perfectly inelastic collision between two freight cars, the momentum of the freight car on the left is shared with the freight car on the right.

In an perfectly inelastic collision, the total kinetic energy does not remain constant when the objects collide and stick together. Some of the kinetic energy is converted to sound energy and internal energy as the objects deform during the collision.

Most collisions usually involve some external force. Billiard balls do not continue indefinitely with the momentum imparted to them. The moving balls encounter friction with the table and the air. These external forces are usually negligible during the collision, so the net momentum does not change during collision. The net momentum of two colliding trucks is the same before and just after the collision. As the combined wreck slides along the pavement, friction provides an impulse to decrease its momentum. Similarly, a pair of space vehicles docking in orbit have the same net momentum just before and just after contact. Since there is no air resistance in space, the combined momentum of the space vehicles after docking is then changed only by gravity.

FIGURE 3. An air track nicely demonstrates conservation of momentum. Many small air jets provide a nearly frictionless cushion of air for the gliders to slide on.

Perfectly elastic collisions are not common in the everyday world. We find in practice that some heat is generated during collisions. Drop a ball and after it bounces from the floor, both the ball and the floor are a bit warmer. Even a dropped superball will not bounce to its initial height. At the microscopic level, however, perfectly elastic collisions are commonplace. For example, electrically charged particles bounce off one another without generating heat; they don’t even touch in the classic sense of the word. Later chapters will show that the concept of touching needs to be considered differently at the atomic level.

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