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

## 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

## Answers and Replies

ideasrule
Homework Helper
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.