A satellite with a mass of 260 kg approaches a large planet at a speed vi,1 = 13.1 km/s. The planet is moving at a speed vi,2 = 10.2 km/s in the opposite direction. The satellite partially orbits the planet and then moves away from the planet in a direction opposite to its original direction (see the figure). If this interaction is assumed to approximate an elastic collision in one dimension, what is the speed of the satellite after the collision? This so-called slingshot effect is often used to accelerate space probes for journeys to distance parts of the solar system
To find the speed of the satellite after the collision, we can use the principle of conservation of momentum in one dimension.
The momentum of an object is defined as the product of its mass and velocity. The total momentum before the collision should be equal to the total momentum after the collision, assuming no external forces are acting on the system.
Let's break down the problem step by step:
Step 1: Convert the given velocities from km/s to m/s (using the conversion factor 1 km/s = 1000 m/s).
- vi,1 = 13.1 km/s = 13,100 m/s
- vi,2 = 10.2 km/s = 10,200 m/s
Step 2: Calculate the momentum before the collision. Since the satellite is moving in the opposite direction to the planet, one of them will have a negative momentum.
- Momentum of the satellite before the collision (p1) = mass of the satellite (m) × velocity of satellite (v1) = 260 kg × (-13,100 m/s)
- Momentum of the planet before the collision (p2) = mass of the planet (M) × velocity of planet (v2) = (assumed to be much larger than satellite mass) × (10,200 m/s)
Step 3: Apply the conservation of momentum principle to find the total momentum after the collision. Since the collision is elastic, the total momentum should remain the same.
- Total momentum before the collision = Total momentum after the collision
- (260 kg × (-13,100 m/s)) + (M × 10,200 m/s) = Total momentum after the collision
Step 4: Solve the equation to determine the total momentum after the collision.
Step 5: Divide the total momentum after the collision by the combined mass of the satellite and planet to find the final velocity of the satellite.
- Final velocity of the satellite (v1,final) = Total momentum after the collision / (mass of the satellite + mass of the planet)
This calculation will give you the speed of the satellite after the collision, which in this case refers to its speed after the slingshot effect.