A \"gravitational slingshot\" is where a spacecraft undergoes an elastic collisi

Question

A "gravitational slingshot" is where a spacecraft undergoes an elastic collision with a planet without touching it, in order to increase its speed and change its direction without using any fuel. The picture to the right shows the planet Saturn moving in the negative x direction at its orbital speed (with respect to the Sun) of vs = -9.5 km/s. The mass of Saturn is 5.69 times 10^26 kg. A spacecraft with a mass of 820 kg approaches Saturn. When far from Saturn it moves in the +x direction at v - 10.3 km/s. The gravitational attraction of Saturn (a conservative force) acting on the spacecraft causes it to swing around the planet and head off in the opposite direction. Estimate the final speed of the spacecraft v' after it is far enough away to be considered free of Saturn's gravitational pull.

Explanation / Answer

The equations to:
let u be velocity before the collision
let v be velocity after the collision
subscripts 1 and 2 denote the spacecraft and planet

v1 = [u1(m1-m2) + 2(m2*u2)] / (m1+m2)
v2 = [u2(m2-m1) + 2(m1*u1)] / (m1+m2)

A little expansion gives:
v1 = [u1(m1-m2) / (m1+m2)] + [2(m2*u2) / (m1+m2)]

As the planet is tremendously more massive than the spacecraft, the values m2, -(m1-m2) and (m1+m2) can be considered equal to many more significant figures than required (note the negative sign). Plugging that into the v1 equation above yields:
v1 = [u1(-m2) / (m2)] + [2(m2*u2) / (m2)]
cancel as appropriate
v1 = -u1 + 2(u2)

Plug in your values
v1 = -(10.3 km/s) + 2(-9.5 km/s)
v1 = -10.3 km/s - 19.0 km/s

v1 = -29.3 km/s ...........Ans.

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