An example of an elastic collision is The gravitational slingshot effect . A spa

Question

An example of an elastic collision is The gravitational slingshot effect . A spacefship of mass 650 kg is moving at 8.0 km/s in the +x direction. It approaches the planet Saturn, mass 5.68 x10 26kg , which is moving in the –x- direction as shown in the figure. The gravitational attraction of Saturn accelerates the spaceship as it approaches and causes to swing around the planet and heads off in the same direction as Saturn. Several probes such as Voyager have used this to save on “gas”.   Estimate the final speed of the spacecraft after it is far enough to be considered free of Saturn’s gravitational pull. Assume that the speed of Saturn is not affect due to its large mass. Express your answer in km/s. HINT: perfectly ELASTIC COLLISION.

Explanation / Answer

The equations boil down 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 = -(10.4 km/s) + 2(-9.6 km/s)
v1 = -10.4 km/s - 19.2 km/s

v1 = -29.6 km/s





As a thought experiment, look at the problem from the reference frame of the planet, so that the planet's velocity appears to be zero. In this case, the spacecraft is approaching the planet at 20 km/s, and will depart the planet at 20 km/s in the other direction (perfectly elastic collision with an immovable object). Now, shift back to a stationary reference by applying the planet's velocity to the whole system.
if u1 = 20 km/s, v1 = -20 km/s
add the planet's velocity to change reference:
v1 = -20 km/s + -9.6 km/s
v1 = -29.6 km/s

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