A spacecraft of mass m1 = 4673 kg with a speed v1i=5.0 ×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=

To find the final speed of the spacecraft after it is far enough away from Saturn to be nearly free of Saturn's gravitational pull, we need to apply the principle of conservation of momentum.

The momentum before the interaction is given by the product of the mass and velocity of the spacecraft:
p1i = m1 * v1i

The momentum of Saturn is given by the product of its mass and velocity:
ps = ms * vs

Since the spacecraft swings around Saturn and heads off in the opposite direction it approached, its final momentum should be equal in magnitude but opposite in direction to its initial momentum:
p1f = -p1i

The total momentum after the interaction is the vector sum of the momentum of the spacecraft and Saturn:
pTotal = p1f + ps

We can solve for the final velocity of the spacecraft (v1f) using the fact that momentum is mass times velocity:
pTotal = m1 * v1f + ms * vs

Since the final speed is defined as the magnitude of the final velocity, we have:
|pTotal| = m1 * |v1f| + ms * |vs|

Substituting the given values into the equation:
|pTotal| = m1 * |v1f| + ms * vs
|pTotal| = 4673 kg * |v1f| + 5.69 × 10^26 kg * 9.6 × 10^3 m/s

Now solve for |v1f| by rearranging the equation:
|v1f| = (|pTotal| - ms * vs) / m1

Substituting the given values:
|v1f| = (|pTotal| - 5.69 × 10^26 kg * 9.6 × 10^3 m/s) / 4673 kg

By plugging in the values for |pTotal|, we can determine the final speed (|v1f|) of the spacecraft after it is far enough away from Saturn to be nearly free of Saturn's gravitational pull.

http://www.learningace.com/doc/2339465/7e7ffe3cc6dd88b45b69cdef6aa9ec29/ic_sol_w07d3-2