Velocity after slingshot at Jupiter

[suspenc3]

Spacecraft voyager 2 (of mass $$m$$ and speed $$v$$ relative to the sun) approaches the planet Jupiter (of mass $$M$$ and speed $$v_J$$ relative to the sun). The spacecraft rounds the planet and departs in the opposite direction. What is its speed, relative to the sun, after this slingshot encounter, which can be anylized as a collision? Assumer $$v = 12km/s$$ and $$v_J = 13km/s$$ (the orbital speed of jupiter). The mass of Jupiter is very much grater than the mass of the spacecraft .

Can someone point me in the right direction?

Thanks

 

[mezarashi]

Remove the fancy terminology and focus on collision mechanics.

Apply the conservation of momentum equation.
Apply conservation of kinetic energy.

Replace M as a function of m, such that all the m's in the momentum equation cancel. Do some factoring and cancelling and you can make the approximation that $$V_J(final) + V_J(initial) \approx 2V_J(initial)$$ since the mass is very very large compared to the space craft.

 

[quicknote]

for your question. I would be happy to provide a response to the content you have provided.

Firstly, let's break down the information we have been given. We know that the spacecraft, Voyager 2, has a mass of m and is traveling at a speed of v relative to the sun. We also know that Jupiter, with a mass of M, is traveling at a speed of v_J relative to the sun.

During the slingshot maneuver, the spacecraft will approach Jupiter and then round the planet, using its gravitational pull to increase its speed. This is because the spacecraft is essentially borrowing some of Jupiter's momentum during the encounter.

To calculate the new velocity of the spacecraft after the slingshot, we can use the conservation of momentum equation: m*v + M*v_J = (m+M)*v'.

Since we know the values for m, M, v, and v_J, we can rearrange the equation to solve for the final velocity, v'. This gives us: v' = (m*v + M*v_J)/(m+M).

Plugging in the values given in the question, we get: v' = (m*12km/s + M*13km/s)/(m+M).

Since the mass of Jupiter (M) is much greater than the mass of the spacecraft (m), we can assume that M >> m. This means that the final velocity, v', will be very close to Jupiter's orbital speed, v_J.

So, the spacecraft's speed, relative to the sun, after the slingshot encounter would be approximately 13km/s, the same as Jupiter's orbital speed.

I hope this helps to point you in the right direction. If you have any further questions, please feel free to ask.