Many spaceprobes utilise the gravitational slingshot effect to increase their speed and hence their kinetic energy. To do this their flight path must pass close to a planet and the probe must be travelling in the opposite direction to the planetin its orbital path about the sun. This interaction obviously cannot be head on but the maximum theoretical increase in speed may be attained for a head on elastic collision. A real spacecraft approaches at an angle.

A diagram should make this clear.

In this page we review the mechanics of a simple elastic collision before going on to apply the theory to the gravitational slingshot. The interested student can then try some sums for spacecraft and planet data.

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Analysis of an Elastic Collision
We consider the case where two masses approach each other head on. We then analyse the motion assuming both linear momentum and kinetic energy are conserved. In the diagram we have shown the motion after the interaction to be in the same direction but this is only for covenience and the treatment will be perfectly general.
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. In a completely elastic collision kinetic energy is conseved. Using the Newtonian form of kinetic energy 1/2mv2 we readily see that for the collision:

m1u12 + m2u22 = m1v12 + m2v22

Conservation of Kinetic Energy

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In addition Linear Momentum is conserved:
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m1u1 + m2u2 = m1v1 + m2v2

Conservation of Linear Momentum

Combining these two equations leads to Newton's Law of collisions (for the case where the Coefficient of Restitution is unity):

- ( u1 - u2 ) = v1 - v2

We are interested for the gravitational slingshot problem in the speed of mass 2 after the collision in terms of the initial speeds of both masses. We need to eliminate v1 from our equations. We have from Newton's Law that

v1 = u2 - u1 + v2

This is substituted into the momentum equation and some rearrangement gives:

For a spaceprobe and planet, the mass of the planet is much larger than the mass of the probe. This equation then simplifies to v2 = 2u1 - u2. Being careful with signs we see that the speed of the probe after the interaction is v2 = 2u1 + u2. The spaceprobe thus gains twice the speed of the planet as well as maintaining its original speed.

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A More Realistic Slingshot
For the situation pictured above, the situation is only changed by the angle of approach to the planet's orbital flightpath. If the collision is to be treated as elastic , the probe must not enter the planet's atmosphere or some kinetic energy would be lost. We must remember the assumptions :

• It is a perfectly elastic collision which conserves kinetic energy.
• The mass of the planet is much greater than the mass of the probe.

Here is a diagram that shows the relevant velocity data.

From the diagram we see that after the interaction the components of the velocity of the spaceprobe are changed as follows:

• The component perpendicular to the motion of the planet is unchanged and is given by Vsinq
• The component parallel to the motion of the planet is now 2U + Vcosq

The speed of the probe relative to the rest of the solar system is now found from the components in the usual way and may be calculated to be

Spaceprobe Speed After Slingshot Interaction

Of course from the point of view of an observer on the planet the spacecraft is moving at a speed of V relative to the planet both before and after the interaction.

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Things To Think About or Try

1. Fill in the missing algebra that leads to Newton's Law of collisions.
2. Derive the formula for the post interaction speed of mass 2 in our analysis of an elastic colision.
3. Derive the spaceprobe speed formula for after the slingshot interaction.
4. The Voyager spaceprobe attained speeds of up to 25km/sec. If it approached the flightpath of Jupiter at 30o then

• (a) Determine the two components of velocity relative to Jupiter's frame of reference.
• (b) If Jupiter orbits the sun every 12 years and is at a mean distance of 8x1011 m from the sun, determine its orbital speed.
• (c) Determine the speed of Voyager after a gravitational slingshot.
• (d) Determine the % increase in the spaceprobe's kinetic energy.