The given diagram illustrates an object of M travelling to the right with velocity 2v colliding with another object of mass 2M travelling to the left with velocity v.
The bodies stick together after the collision, What are the total momentum and the total kinetic energy of the 2 objects after the collision?
To find the total momentum and kinetic energy of the two objects after the collision, we need to consider the conservation of momentum and the conservation of kinetic energy.
1. Conservation of momentum:
According to the law of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.
The initial momentum of the first object (M) traveling to the right with velocity 2v is given by:
P1(initial) = mass (m1) x velocity (v1)
P1(initial) = M x 2v
P1(initial) = 2Mv
The initial momentum of the second object (2M) traveling to the left with velocity v is given by:
P2(initial) = mass (m2) x velocity (v2)
P2(initial) = 2M x (-v)
P2(initial) = -2Mv
Since the two objects stick together after the collision, their final combined momentum will be equal to the total initial momentum:
P(final) = P1(initial) + P2(initial)
P(final) = 2Mv + (-2Mv)
P(final) = 0
So, the total momentum of the two objects after the collision is zero.
2. Conservation of kinetic energy:
The law of conservation of kinetic energy states that the total kinetic energy before the collision is equal to the total kinetic energy after the collision.
The initial kinetic energy of the first object (M) is given by:
KE1(initial) = (1/2) x mass (m1) x velocity (v1)^2
KE1(initial) = (1/2) x M x (2v)^2
KE1(initial) = 2Mv^2
The initial kinetic energy of the second object (2M) is given by:
KE2(initial) = (1/2) x mass (m2) x velocity (v2)^2
KE2(initial) = (1/2) x 2M x v^2
KE2(initial) = Mv^2
Since the two objects stick together after the collision, their final combined kinetic energy will be equal to the total initial kinetic energy:
KE(final) = KE1(initial) + KE2(initial)
KE(final) = 2Mv^2 + Mv^2
KE(final) = 3Mv^2
So, the total kinetic energy of the two objects after the collision is 3Mv^2.