How Does a Rocket's Trajectory Change Near a Star?

[skizm240]

A rocket is 1.00x10^10m from a moving star moving at a speed of 1.00x10^5 m/s with a total energy of -1.67x10^17 J when it experiences a gravitational force magnitude of 6.67x10^7 N. It turns on its thrusters and its speed as well its distance to the star doubles by the time the thrusters are turned off.

Find the rocket's mass, the star's mass
Prove the fist trajectory is elliptical and not circular.
If a=1.11x10^10 m for the first trajectory, find the period.
Find the work done by the thrusters in changing trajectories.

How would I star this problem? It is the first part of physics ( Mech)

I know r'= 2r and v'=2v
would I use the orbtial formulas to find the mass or something else. Any hints or anything will be helpful thank you.

 

[SpaceTiger]

Find the rocket's mass, the star's mass

What's the expression for the total energy of an object in orbit about a central mass? What's the expression for the gravitational force between two objects?

Prove the fist trajectory is elliptical and not circular.

What's the velocity of an object in a circular orbit at this distance? How does it compare to the rocket's velocity?

If a=1.11x10^10 m for the first trajectory, find the period.

What is Kepler's Third Law?

Find the work done by the thrusters in changing trajectories.

What is the energy of the new orbit? Where did this energy come from?

 

[quicknote]

To start this problem, you would first need to identify the given values and variables. The given values are the rocket's initial distance from the star (1.00x10^10 m), the star's initial speed (1.00x10^5 m/s), the total energy of the system (-1.67x10^17 J), and the gravitational force magnitude (6.67x10^7 N). The variables are the rocket's mass (m1), the star's mass (m2), the rocket's final distance from the star (2.00x10^10 m), and the rocket's final speed (2.00x10^5 m/s).

To find the rocket's mass and the star's mass, you can use the equations for gravitational potential energy and kinetic energy:

Total energy = gravitational potential energy + kinetic energy

-1.67x10^17 J = (-6.67x10^7 N)(m1)(m2)/1.00x10^10 m + (0.5)(m1)(2.00x10^5 m/s)^2

Solving for m1 and m2, you can find the masses of the rocket and the star.

To prove that the first trajectory is elliptical and not circular, you can use the formula for eccentricity:

e = (r_max - r_min)/(r_max + r_min)

For a circular orbit, the eccentricity is 0. Therefore, if the calculated eccentricity for the first trajectory is not 0, it can be concluded that the trajectory is elliptical.

To find the period for the first trajectory, you can use the equation for orbital period:

T = 2π√(a^3/G(m1+m2))

Where a is the semi-major axis, which can be calculated using the given value of a=1.11x10^10 m.

To find the work done by the thrusters in changing trajectories, you can use the formula for work:

W = ΔK = K_final - K_initial

Where K is the kinetic energy of the rocket, which can be calculated using the final and initial speeds of the rocket.

You can also use the equations for centripetal force and centripetal acceleration to solve this problem. Remember to convert all given values to SI units before using the equations.