Change orbit from circle to parabola

[athrun200]

1(a) Homework Statement
A spaceship travels in a circular orbit around a planet. It applies a sudden
thrust and increases its speed by a factor f . If the goal is to change the
orbit from a circle to a parabola, what should f be if the thrust points in the tangential direction?

1(b) Is your answer any different if the thrust points
in some other direction? What is the distance of closest approach if the
thrust points in the radial direction?

Homework Equations

Kepler’s laws

The Attempt at a Solution

We know that for circle, ε=0; for parabola ε=1
We also know that $ε=\sqrt{1+\frac{2EL^{2}}{m\alpha^{2}}}$
So $E=-\frac{m\alpha^{2}}{2L^{2}}$ for ε=0
$E=0$ for ε=1

But I don't know what to do next in order to find the increased factor f

 

[TSny]

Hello. Could you please define the symbols $\alpha$ and $L$?

Can you express the kinetic energy of the ship in terms of $m$, $\alpha$ and $L$ before and after the thrust?

 

[athrun200]

Hello. Could you please define the symbols $\alpha$ and $L$?

Can you express the kinetic energy of the ship in terms of $m$, $\alpha$ and $L$ before and after the thrust?

$V(r)=-\frac{\alpha}{r}$
So $\alpha=GMm$

$L$ is angular momentum.

I think it is possible to express KE in terms of m and L but not $\alpha$.
But what should I do next?

 

[TSny]

$V(r)=-\frac{\alpha}{r}$
So $\alpha=GMm$

$L$ is angular momentum.

I think it is possible to express KE in terms of m and L but not $\alpha$.
But what should I do next?

ok. Express kinetic energy in terms of $\alpha$ and $r$ using E= KE + V
You can assume $r$ remains constant during the thrust. So you should be able to relate initial and final speeds since you know how E changes. Instead of expressing E in terms of L, try to express E in terms of r for a circular orbit.

 

[paranormal]

. Can you please provide some guidance on how to approach this problem?

To change the orbit from a circle to a parabola, the specific energy (E) of the spaceship needs to increase from 0 to 0.5mα^2. This means that the thrust needs to provide enough kinetic energy to increase the speed of the spaceship by a factor of f that satisfies the equation:

E= -\frac{m\alpha^2}{2L^2} * (f^2 - 1)

If the thrust points in the tangential direction, the angular momentum (L) remains constant and the equation simplifies to:

E= -\frac{m\alpha^2}{2L^2} * (f^2 - 1) = -\frac{m\alpha^2}{2L^2} * (f^2 - 0) = -\frac{m\alpha^2}{2L^2} * f^2

To find the value of f, we can rearrange the equation to solve for f:

f=\sqrt{\frac{2EL^2}{m\alpha^2}}=\sqrt{\frac{2*0.5m\alpha^2}{m\alpha^2}}=\sqrt{1}=1

Therefore, the factor f needs to be 1 in order to change the orbit from a circle to a parabola when the thrust points in the tangential direction.

If the thrust points in any other direction, the angular momentum (L) will change and the equation becomes more complicated. The distance of closest approach would also depend on the direction of the thrust and the initial position and velocity of the spaceship. Without more information, it is difficult to determine the exact value of f or the distance of closest approach in this case.