# Orbital Mechanics - effect of thrust on v, R, a

• awygle
Similarly for b), exerting a force anti-parallel to the velocity vector decreases v but doesn't immediately change R. For c) and d), the change in force results in a change in both v and R, but the specific effects on each quantity depend on the direction of the force relative to the velocity and position vectors. In summary, the effects of the spaceship's engines on the quantities a_{c}, v^{2}, and R depend on the direction of the force exerted and can result in changes to both v and R.

## Homework Statement

A spacecraft orbits around the Earth at a tangential speed v and a radius R, in accordance with the equation $$a_{c} = \frac{v^{2}}{R}$$ .

a) If the spaceship's engines exert a force parallel to its instantaneous vector of motion, what is the effect on the quantities $$a_{c}, v^{2}$$, and $$R$$?
b) If the spaceship's engines exert a force anti-parallel to its instantaneous vector of motion, what is the effect on the quantities $$a_{c}, v^{2}$$, and $$R$$?
c) If the spaceship's engines exert a force perpendicular to its instantaneous vector of motion and anti-parallel to its position vector (towards the planet), what is the effect on the quantities $$a_{c}, v^{2}$$, and $$R$$?
d) If the spaceship's engines exert a force perpendicular to its instantaneous vector of motion and parallel to its position vector (away from the planet), what is the effect on the quantities $$a_{c}, v^{2}$$, and $$R$$?

## Homework Equations

$$a_{c} = \frac{v^{2}}{R}$$
Various rotational kinematics equations (?)

## The Attempt at a Solution

It's asking whether the quantities change, and if so in what direction. For a) I got that since a is roughly a constant, increasing v must increase R, and for b) a is roughly constant, decreasing v must decrease R. But I am confused about c) and d). Changes in either R or v, or both, could balance the equation with the change in a created by the thrust. So for the homework I'm wondering which would change, and for my own personal knowledge I'm wondering by how much (or at least generally how to compute that).

Thanks

If the engines are fired when the spacecraft is in a circular orbit, the spacecraft MUST end up in an elliptical orbit. That's why I suspect the questions refer to the instant after the engines are fired. So for example, for a), exerting a force parallel to the velocity vector doesn't immediately change R, but does increase v.