A spacecraft of mass = 4673 kg with a speed 5.0 10 m/s approaches Saturn which is moving in the opposite direction with a speed 9.6 10 m/s. After interacting gravitationally with Saturn, the spacecraft swings around Saturn and heads off in the opposite direction it approached. The mass of Saturn is = 5.69 10 kg. Find the final speed (in m/s) of the spacecraft after it is far enough away from Saturn to be nearly free of Saturn's gravitational pull.
To find the final speed of the spacecraft, we can use the principle of conservation of momentum.
The momentum of the spacecraft before interacting with Saturn can be calculated by multiplying its mass (m1 = 4673 kg) by its initial velocity (v1 = 5.0 × 10^3 m/s):
Momentum_before = m1 * v1
Similarly, the momentum of Saturn can be calculated by multiplying its mass (m2 = 5.69 × 10^24 kg) by its velocity (v2 = -9.6 × 10^3 m/s), considering the opposite direction:
Momentum_Saturn = m2 * v2
Since momentum is conserved in this system, the total momentum before interacting should be equal to the total momentum after interacting. Therefore, we have:
Momentum_before + Momentum_Saturn = Momentum_after
Now, let's calculate the momentum after interacting. The momentum of the spacecraft after interacting with Saturn can be calculated by multiplying its mass (m1) by its final velocity (v1_f). Since the spacecraft swings around Saturn and heads in the opposite direction, the final velocity (v1_f) will be negative:
Momentum_after = m1 * v1_f
Substituting the values:
Momentum_before + Momentum_Saturn = Momentum_after
m1 * v1 + m2 * v2 = m1 * v1_f
Now, we can solve for the final velocity (v1_f):
v1_f = (m1 * v1 + m2 * v2) / m1
Substituting the given values:
v1_f = (4673 kg * 5.0 × 10^3 m/s + 5.69 × 10^24 kg * (-9.6 × 10^3 m/s)) / 4673 kg
Calculating the final velocity:
v1_f ≈ -9.734 × 10^3 m/s
So, the final speed of the spacecraft after it is far enough away from Saturn to be nearly free of Saturn's gravitational pull is approximately 9.734 × 10^3 m/s in the opposite direction.