NASA, when it sends probes to other planets, uses what is known as a gravitational slingshot. This is when a probe uses the gravitational potential energy a planet to gain kinetic energy. Even though the probe and the planet are not physically colliding, one can treat this problem as a perfectly elastic head on collision between the probe and the planet. Suppose the probe, with a mass of one millionth the mass of the planet, is approaching the planet initially at 269.8 m/s in the negative x direction and the planet is moving at 31.4 m/s in the positive x direction. What is the magnitude of the final velocity of the probe, in m/s, after the collision?
Please see the first Related Question below.
To solve this problem, we can use the concept of conservation of momentum. Before the collision, the total momentum in the x-direction is given by the sum of the momentum of the probe and the momentum of the planet.
Momentum of the probe = mass of the probe × velocity of the probe
Momentum of the planet = mass of the planet × velocity of the planet
Using the given values, we can calculate the initial momentum in the x-direction:
Initial momentum = (mass of the probe × velocity of the probe) + (mass of the planet × velocity of the planet)
= (1 × 10^-6 kg)(-269.8 m/s) + (1 kg)(31.4 m/s)
Next, we need to consider the collision as a perfectly elastic head-on collision, where the total momentum in the x-direction is conserved. After the collision, the probe and the planet will have new velocities in the x-direction, but the total momentum will remain the same.
Therefore, the final momentum in the x-direction is equal to the initial momentum:
Final momentum = Initial momentum
Now, let v_p be the final velocity of the probe and v_pl be the final velocity of the planet after the collision. We know that the mass of the probe is one millionth the mass of the planet.
Final momentum = (mass of the probe × final velocity of the probe) + (mass of the planet × final velocity of the planet)
= (1 × 10^-6 kg)(v_p) + (1 kg)(v_pl)
Since the total momentum in the x-direction is conserved:
Initial momentum = Final momentum
(mass of the probe × velocity of the probe) + (mass of the planet × velocity of the planet) = (mass of the probe × final velocity of the probe) + (mass of the planet × final velocity of the planet)
Substituting the values we know:
(1 × 10^-6 kg)(-269.8 m/s) + (1 kg)(31.4 m/s) = (1 × 10^-6 kg)(v_p) + (1 kg)(v_pl)
Now, we solve the equation to find the values of v_p and v_pl.