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.
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.
In addition Linear Momentum is conserved:
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
v1 = u2 - u1 + v2
This is substituted into the momentum equation and some rearrangement gives:
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.
- 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
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.
Things To Think About or Try
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.