A 25.1 g marble moving to the right at 22.9 cm/s overtakes and collides elastically with a 12.1 g marble moving in the same direction at 11.3 cm/s. After the collision, the 12.1 g marble moves to the right at 25.1 cm/s. Find the velocity of the 25.1 g marble after the collision
To find the velocity of the 25.1 g marble after the collision, we can use the principles of conservation of momentum and kinetic energy in an elastic collision.
Let's denote the velocity of the 25.1 g marble before the collision as v1 and the velocity of the 12.1 g marble before the collision as v2.
According to conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.
The momentum of an object is given by the product of its mass and velocity. Therefore, the momentum of the 25.1 g marble before the collision is (25.1 g) * (22.9 cm/s), and the momentum of the 12.1 g marble before the collision is (12.1 g) * (11.3 cm/s).
Since the marbles are moving in the same direction, the momentum of the system before the collision is given by:
(25.1 g) * (22.9 cm/s) + (12.1 g) * (11.3 cm/s)
According to conservation of momentum, the total momentum after the collision is the sum of the momentum of the 25.1 g marble and the momentum of the 12.1 g marble after the collision.
Let's denote the velocity of the 25.1 g marble after the collision as v1' and the velocity of the 12.1 g marble after the collision as v2'.
Therefore, the total momentum after the collision is:
(25.1 g) * (v1') + (12.1 g) * (25.1 cm/s).
Since it is an elastic collision, kinetic energy is also conserved. The kinetic energy before the collision is the sum of the kinetic energy of the 25.1 g marble and the kinetic energy of the 12.1 g marble before the collision.
The kinetic energy is given by the formula:
(1/2) * m * v^2
Therefore, the total kinetic energy before the collision is:
(1/2) * (25.1 g) * (22.9 cm/s)^2 + (1/2) * (12.1 g) * (11.3 cm/s)^2.
Similarly, the total kinetic energy after the collision is:
(1/2) * (25.1 g) * (v1')^2 + (1/2) * (12.1 g) * (25.1 cm/s)^2.
Now, we have two equations based on conservation of momentum and conservation of kinetic energy. We can solve these two equations simultaneously to find the velocity of the 25.1 g marble after the collision (v1').
Solving the equations will give us the answer.