9.10
Elastic Collisions: Case Study
Elastic collision of a system demands conservation of both momentum and kinetic energy. To solve problems involving one-dimensional elastic collisions…
Suppose marble A and B undergo one-dimensional collision. From the conservation of momentum and kinetic energy, the equation of elastic collision can be written.
If marble B is initially at rest, then the equations get simplified. On solving the two equations, the final velocity of the marbles A and B can be obtained.
If marbles of equal masses collide, then marble A comes to rest after the collision and B travels with the initial velocity of marble A. By solving the equations, it is seen that marbles exchange momentum.
If marble B is heavier than marble A, then, after collision, marble A bounces back with almost the same velocity, and marble B moves with a very low velocity.
If marble A is heavier than marble B, then marble A continues to move with the same velocity. Marble B gets a push and travels with a higher velocity than the initial velocity of marble A.
In the case of marbles of different masses undergoing one-dimensional elastic collisions, the relative velocities before and after the collision have the same magnitude but opposite direction.
View the full transcript and gain access to JoVE Core videos
Q1: What two principles must be satisfied in an elastic collision?
An elastic collision must satisfy conservation of momentum and conservation of kinetic energy. The sum of momentum before the collision equals the total momentum after the collision. Similarly, the sum of kinetic energies before the collision equals the sum after the collision. These two equations allow you to solve for unknown final velocities.
Q2: What happens when two marbles of equal mass collide elastically?
When marbles of equal mass collide elastically, they exchange momentum. The moving marble comes to rest after the collision, while the initially stationary marble travels with the initial velocity of the first marble. This momentum exchange is a direct result of solving the conservation equations for equal-mass objects.
Q3: How does a lighter object behave when it collides with a heavier stationary object?
When a lighter object collides elastically with a heavier stationary object, the lighter object bounces backward with nearly the same velocity it had before collision. The heavier object moves forward but with very low velocity. This outcome is intuitive: a compact car bouncing backward off a stationary full-size SUV demonstrates this principle.
Q4: What is the result when a heavier object strikes a lighter stationary object?
When a heavier object collides elastically with a lighter stationary object, the heavier object continues moving with approximately the same velocity. The lighter object receives a push and travels with a higher velocity than the initial velocity of the heavier object. The lighter object gains significantly more speed due to the mass difference.
Q5: How do relative velocities change before and after an elastic collision?
In one-dimensional elastic collisions between objects of different masses, the relative velocities before and after collision have the same magnitude but opposite directions. This relationship holds regardless of the mass ratio and is a fundamental property derived from the conservation equations for elastic collisions.
Q6: How can you find final velocities in a one-dimensional elastic collision?
To find final velocities, write the conservation of momentum and conservation of kinetic energy equations, then solve them simultaneously. When one object is initially at rest, the equations simplify significantly. Substituting known values like masses and initial velocities yields the two unknown final velocities algebraically.
Q7: Why does a small object bouncing off a larger stationary object move backward?
A small object bounces backward because the larger stationary object exerts a greater force during collision. The conservation equations show that when a lighter object strikes a much heavier object at rest, the lighter object's velocity reverses direction. This negative final velocity indicates backward motion after the collision.