Two manned satellites approaching one another at a relative speed of 0.200 m/s intend to dock. The first has a mass of 4.50 multiplied by 103 kg, and the second a mass of 7.50 multiplied by 103 kg. Assume that the positive direction is directed from the second satellite towards the first satellite.
(a) Calculate the final velocity after docking, in the frame of reference in which the first satellite was originally at rest.
(b) What is the loss of kinetic energy in this inelastic collision?
(c) Repeat both parts, in the frame of reference in which the second satellite was originally at rest.
Find the final velocity and loss of kinetic energy
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To solve this problem, we need to apply the law of conservation of momentum and the law of conservation of kinetic energy.
(a) In the frame of reference in which the first satellite was originally at rest, we can use the law of conservation of momentum to find the final velocity of the system after docking.
The law of conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision.
Before docking, the momentum of the first satellite is zero since it is initially at rest. The momentum of the second satellite is given by:
momentum = mass * velocity = (7.50 * 10^3 kg) * (-0.200 m/s) = -1500 kg·m/s (negative because it is directed towards the first satellite)
After docking, the two satellites will move together, so their final total momentum will be the sum of their individual momenta:
total momentum = (4.50 * 10^3 kg) * final velocity + (7.50 * 10^3 kg) * final velocity
Since this momentum must be conserved, we can set it equal to the initial momentum:
0 + (-1500 kg·m/s) = (4.50 * 10^3 kg + 7.50 * 10^3 kg) * final velocity
Simplifying the equation gives:
-1500 kg·m/s = (12 * 10^3 kg) * final velocity
Dividing both sides by 12 * 10^3 kg:
final velocity = -1500 kg·m/s / (12 * 10^3 kg) = -0.125 m/s
Therefore, the final velocity after docking, in the frame of reference in which the first satellite was originally at rest, is -0.125 m/s (directed towards the first satellite).
(b) To find the loss of kinetic energy in this inelastic collision, we can use the law of 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.
Before docking, the kinetic energy of the second satellite is given by:
kinetic energy = (1/2) * mass * velocity^2 = (1/2) * (7.50 * 10^3 kg) * (-0.200 m/s)^2 = 150 J
After docking, the two satellites will move together with a final velocity of -0.125 m/s, as calculated in part (a). The total kinetic energy after docking is given by:
total kinetic energy = (1/2) * (4.50 * 10^3 kg + 7.50 * 10^3 kg) * (-0.125 m/s)^2
Simplifying the equation gives:
total kinetic energy = (12 * 10^3 kg) * (0.0156 m^2/s^2) = 187.2 J
The loss of kinetic energy is the difference between the initial and final kinetic energies:
loss of kinetic energy = initial kinetic energy - total kinetic energy
= 150 J - 187.2 J
= -37.2 J
≈ -37 J (to two significant figures)
Therefore, the loss of kinetic energy in this inelastic collision is approximately -37 J.
(c) To solve part (c) in the frame of reference in which the second satellite was originally at rest, we can use the same approach as in part (a), but with the velocities switched.
The momentum of the first satellite before docking is given by:
momentum = mass * velocity = (4.50 * 10^3 kg) * (0.200 m/s) = 900 kg·m/s
After docking, the two satellites will move together, so their final total momentum will be the sum of their individual momenta:
total momentum = (4.50 * 10^3 kg) * final velocity + (7.50 * 10^3 kg) * final velocity
Setting the total momentum equal to the initial momentum gives:
900 kg·m/s = (4.50 * 10^3 kg + 7.50 * 10^3 kg) * final velocity
Simplifying the equation gives:
final velocity = 900 kg·m/s / (12 * 10^3 kg) = 0.075 m/s
Therefore, the final velocity after docking, in the frame of reference in which the second satellite was originally at rest, is 0.075 m/s.
To find the loss of kinetic energy, we can use the same approach as in part (b). The initial kinetic energy of the first satellite is given by:
kinetic energy = (1/2) * mass * velocity^2 = (1/2) * (4.50 * 10^3 kg) * (0.200 m/s)^2 = 90 J
The total kinetic energy after docking is given by:
total kinetic energy = (1/2) * (4.50 * 10^3 kg + 7.50 * 10^3 kg) * (0.075 m/s)^2
Simplifying the equation gives:
total kinetic energy = (12 * 10^3 kg) * (0.00281 m^2/s^2) = 33.72 J
The loss of kinetic energy is the difference between the initial and final kinetic energies:
loss of kinetic energy = initial kinetic energy - total kinetic energy
= 90 J - 33.72 J
= 56.28 J
≈ 56 J (to two significant figures)
Therefore, the loss of kinetic energy in this inelastic collision, in the frame of reference in which the second satellite was originally at rest, is approximately 56 J.