THE
GRAVITATIONAL SLINGSHOT
A An
Application of Newtonian Mechanics
.
Gravitation Main
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.
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 a completely elastic
collision kinetic energy is conseved. Using the Newtonian form of kinetic
energy 1/2mv^{2 }we readily see that for the collision:
.
Combining these two
equations leads to Newton's Law of collisions
(for the case where the Coefficient of Restitution is unity):
.
- ( u_{1}
- u_{2} ) = v_{1 }- v_{2} _{.}
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 v_{1
}from
our equations. We have from Newton's Law that
v_{1
}=
u_{2 }- u_{1 }+ v_{2} _{.}
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 v_{2}
= 2u_{1} - u_{2}. Being
careful with signs we see that the speed of the probe after the interaction
is v_{2} = 2u_{1}
+ u_{2}. The spaceprobe thus
gains twice the speed of the planet as well as maintaining its original
speed.
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.
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 30^{o} 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 8x10^{11} 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.