One sphere of mass 1 kg is moving at 5 m/s to the right until it collides with a stationary, 2 kg sphere. After the collision, both spheres travel to the right: the 1 kg sphere at 1 m/s, and the 2 kg sphere at 2 m/s. What kind of collision took place?
To determine what kind of collision took place, we need to consider the conservation of momentum and kinetic energy.
The initial momentum of the system is given by the sum of individual momenta:
Initial momentum = (mass of 1 kg sphere) * (velocity of 1 kg sphere) + (mass of 2 kg sphere) * (velocity of 2 kg sphere)
Initial momentum = (1 kg) * (5 m/s) + (2 kg) * (0 m/s)
Initial momentum = 5 kg·m/s
The final momentum of the system can be calculated in the same way:
Final momentum = (mass of 1 kg sphere) * (velocity of 1 kg sphere) + (mass of 2 kg sphere) * (velocity of 2 kg sphere)
Final momentum = (1 kg) * (1 m/s) + (2 kg) * (2 m/s)
Final momentum = 5 kg·m/s
Since the initial and final momenta are the same, we have momentum conservation. This indicates that an elastic collision took place.
To check if the collision is truly elastic, we can look at the conservation of kinetic energy. The initial kinetic energy is given by the sum of individual kinetic energies:
Initial kinetic energy = 0.5 * (mass of 1 kg sphere) * (velocity of 1 kg sphere)^2 + 0.5 * (mass of 2 kg sphere) * (velocity of 2 kg sphere)^2
Initial kinetic energy = 0.5 * (1 kg) * (5 m/s)^2 + 0.5 * (2 kg) * (0 m/s)^2
Initial kinetic energy = 25 J
The final kinetic energy is given by the sum of individual kinetic energies:
Final kinetic energy = 0.5 * (mass of 1 kg sphere) * (velocity of 1 kg sphere)^2 + 0.5 * (mass of 2 kg sphere) * (velocity of 2 kg sphere)^2
Final kinetic energy = 0.5 * (1 kg) * (1 m/s)^2 + 0.5 * (2 kg) * (2 m/s)^2
Final kinetic energy = 2.5 J
Since the initial and final kinetic energies are different, the collision is not perfectly elastic. It is a partially elastic collision.