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

AI Thread Summary
To calculate the slingshot maneuver dynamics for a space probe, specific energy, angular momentum, and closest approach distance must be determined using gravitational principles. The mass of the planet and probe, along with their relative speed and impact parameter, are essential for these calculations. The equations of conservation of energy and angular momentum are crucial, but the absence of Rmax complicates finding the semi-major axis and eccentricity. The closest approach distance allows for the calculation of maximum speed using these conservation principles. Understanding these dynamics is vital for successfully navigating the probe's trajectory.
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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
 
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The only equations necessary here are those of conservation of energy and angular momentum.
 
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.
 
Once you know (2), you can use both conservation of energy and conservation of angular momentum to obtain the max speed.
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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