How Do You Calculate Slingshot Maneuver Dynamics for a Space Probe?

[noreally]

Really need help with this past exam question for revison:

Homework Statement

A probe performs slingshot maoeuvre around planet. Mass of planet is 9x10^22kg. Mass of probe is 4x10^2 kg and approaches planet from a great distance with relative speed 2x10^3 m/s and impact parameter 6x10^7. Using gravitational constant, G, determine:

1) Specific Energy, C and Specific Angular momentum L and alpha.
2) Closest approach distance of probe
3) Max speed of probe

I have the equations, but they involve the semi-major axis which can't be used as Rmax and Rmin arent given here so I'm a little lost on where to being with this, if anyone could help out I would be eternally greatfull! :D

 

[voko]

The only equations necessary here are those of conservation of energy and angular momentum.

 

[noreally]

I found parts 1 and 2 but am stuck on finding Vmax at my Rmin value without knowing the Rmax to be able to calculate the semi-major axis and eccentricity.

 

[voko]

Once you know (2), you can use both conservation of energy and conservation of angular momentum to obtain the max speed.

 

[HalfLostAndSearching]

Probe slingshot dynamics is a fascinating and complex topic in space exploration. In order to answer this past exam question, we first need to understand the basic principles of slingshot maneuvers.

A slingshot maneuver, also known as a gravity assist or a swing-by, is a technique used by spacecraft to gain speed and change direction by using the gravitational pull of a planet or other celestial body. This allows the spacecraft to conserve fuel and reach its destination more efficiently.

Now, let's look at the given information. We have the mass of the planet, the mass of the probe, the relative speed of the probe, and the impact parameter. Using the gravitational constant, G, we can calculate the specific energy, specific angular momentum, and alpha for the probe.

1) Specific Energy, C, is given by the equation:

C = -GM/2a

Where G is the gravitational constant, M is the mass of the planet, and a is the semi-major axis of the probe's orbit. However, as you have correctly pointed out, we do not have the semi-major axis. In this case, we can use the fact that the specific energy remains constant throughout the slingshot maneuver. Thus, we can use the given relative speed and impact parameter to calculate the specific energy as follows:

C = v^2/2 - GM/b

Where v is the relative speed and b is the impact parameter.

Specific Angular Momentum, L, is given by the equation:

L = rbv

Where r is the distance between the probe and the planet, b is the impact parameter, and v is the relative speed. Using the given information, we can calculate the specific angular momentum.

Alpha, α, is defined as the angle between the incoming and outgoing trajectories of the probe. It can be calculated using the following equation:

α = sin^-1 (b/rc)

Where r is the distance between the probe and the planet, b is the impact parameter, and c is the closest approach distance.

2) The closest approach distance, c, can be calculated using the fact that the specific energy remains constant. Thus, we can use the given specific energy and impact parameter to calculate the closest approach distance as follows:

c = GM/v^2 - b

3) The maximum speed of the probe, vmax, can be calculated using the equation:

vmax = √(2GM/b)

Where G is the gravitational constant, M is the mass of