A spacecraft of mass m1 = 2161 kg with a speed v1i=7.0 ×103 m/s approaches Saturn which is moving in the opposite direction with a speed vs=9.6×103 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 ms = 5.69 × 1026 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.
v1f=
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To find the final speed of the spacecraft (v1f), we can use the principle of conservation of momentum. The total momentum of the system before the interaction is equal to the total momentum after the interaction.
Before the interaction, the momentum of the spacecraft is given by:
p1i = m1 * v1i
The momentum of Saturn is given by:
ps = ms * vs
After the interaction, the spacecraft changes direction and moves in the opposite direction with a final velocity v1f. The momentum of the spacecraft is then given by:
p1f = m1 * v1f
The momentum of Saturn does not change, as it continues moving in the opposite direction with a constant velocity. Therefore, the total momentum after the interaction is:
p_total = p1f + ps
Using the conservation of momentum, we can set the initial momentum equal to the final momentum:
p1i = p_total
m1 * v1i = m1 * v1f + ms * vs
Now we can solve for v1f:
v1f = (m1 * v1i + ms * vs) / m1
Plugging in the given values:
v1f = (2161 kg * 7.0 × 10^3 m/s + 5.69 × 10^26 kg * 9.6 × 10^3 m/s) / 2161 kg
Calculating this expression will give us the final speed (v1f) of the spacecraft after it is far enough away from Saturn to be nearly free of Saturn's gravitational pull.