Car 1 rolls at a constant speed and collides elastically with car 2, which has the same mass but is at rest. The collision brings car 1 to rest. How does the speed of car 2 after the collision compare with the speed of car 1 at collision?
same momentum before and after
In an elastic collision, both kinetic energy and momentum are conserved. Let's analyze the scenario step-by-step to determine the relationship between the speeds of car 1 and car 2 after the collision:
Step 1: Before the collision
Car 1 is rolling at a constant speed, which means it has momentum (mass × velocity). Car 2 is at rest, so its momentum is zero.
Step 2: During the collision
When car 1 collides with car 2, the collision is elastic, which means both kinetic energy and momentum are conserved. The momentum of car 1 is transferred to car 2.
Step 3: After the collision
Since car 1 comes to rest after the collision, its final velocity is zero. Car 2 gains the momentum from car 1 and starts moving. Both cars have equal masses, so the magnitude of the momentum transferred to car 2 is the same as the momentum of car 1 before the collision.
Based on the conservation of momentum, the speed of car 2 after the collision will be the same as the speed of car 1 before the collision.
Therefore, the speed of car 2 after the collision is equal to the speed of car 1 at the collision.
To determine the speed of car 2 after the collision, we can use the principles of conservation of momentum and kinetic energy in an elastic collision.
The conservation of momentum states that the total momentum of the system before the collision is equal to the total momentum after the collision. In this case, car 2 is initially at rest, so its momentum is zero. Hence, the total momentum before the collision is equal to the momentum of car 1.
The conservation of kinetic energy states that the total kinetic energy of the system before the collision is equal to the total kinetic energy after the collision. Since car 1 comes to rest after the collision, its kinetic energy is zero. Therefore, the total kinetic energy after the collision is equal to the kinetic energy of car 2.
Let's denote the initial velocity of car 1 as v1 and the final velocity of car 2 as v2.
Using the conservation of momentum:
(mass of car 1)(v1) = (mass of car 2)(v2)
Since both cars have the same mass:
v1 = v2
Therefore, the speed of car 2 after the collision is equal to the speed of car 1 before the collision.