Yet another problem from classical mechanics

[haplo]

Homework Statement

THis time it is rather 3 problems, you are required to solve two of them before tackling third.

So, in the first problem is about hoffman transfe. YOu are asked to calculate the velocity of the spaceship when it leaves the Earth and at it's encounter with juputer. It was easily done through conservation f angular momentum and energy equations. Velocity of spaceship is 7.7km/s at jupiters orbit and juputer is 13.1 So the velocity relative to Jupiter is 5.4.

The second problem is about slingshot manuever :

the problems states: ON reaching the vicinity of Jupiter the spaaroundcecraft is swung around the planet by it's gravitational attraction. Consider the encounter in a frame of reference in which Jupiter is at rest. What is magnitude and direction of the spacecraft s velocity before scattering and magnitude after scattering. If scattering angle in this frame is 90 degress what is it's impact parameter and distance of closest approach.

THis one was also easy to solve. From previous problem we know that velocity relative to Jupiter is 7.4 in sun refference amd direction is the same as juputers. So in juputers frame it is 5.4 towards juputer. After scattering the velocity should be the same 5.4. For second part impact paramter is easily calculated from b=a*cot(1/2*theta), where a=(Rjupiter+Rearth)/2. And distance of closest approach from radial energy equation by setting Vr=0 and calculating R.

And here comes the last problem.
If the manuever described in previos problem is in orbital plane so that final velocity of the spacecraft is radially away from the sun. What is it's magnitude and direction relative to the sun. Use radial energy equation to determine spacecraft s furthers distance from the sun. FInd new orbital period.

Any hints on how to complete the first part. It looks difficult to understand what impact paramter should be as well as scattering angel? What values from previous problems should you use?

Homework Equations

The Attempt at a Solution

 

[Jhero]

Hello fellow forum member, thank you for posting these interesting problems. I am a scientist and I would like to offer some assistance in solving the last problem.

Firstly, let's review the information we have from the previous two problems. We know that the velocity of the spacecraft relative to Jupiter is 5.4 km/s and the direction is towards Jupiter. We also know that the spacecraft's velocity before and after scattering is 5.4 km/s. From the second problem, we calculated the impact parameter and the distance of closest approach using the formula b=a*cot(1/2*theta) and the radial energy equation.

Now, for the third problem, we need to consider the orbital plane of the spacecraft. This means that the final velocity of the spacecraft should be in the same plane as its orbit around the sun. Since the velocity after scattering is radially away from the sun, we can use the law of conservation of angular momentum to determine the magnitude and direction of the spacecraft's velocity relative to the sun.

We can use the equation L=mrv, where L is the angular momentum, m is the mass of the spacecraft, r is the distance from the sun, and v is the velocity. Since we know the initial and final velocities, we can solve for r. Once we have the distance, we can use the radial energy equation to determine the spacecraft's furthest distance from the sun.

To find the new orbital period, we can use Kepler's third law, which states that the square of the orbital period is proportional to the cube of the semi-major axis. We can use the semi-major axis we calculated from the previous problem and the new distance from the sun to determine the new orbital period.

I hope this helps guide you in solving the last problem. If you have any further questions or need clarification, please don't hesitate to ask. Good luck!