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A satellite is in a circular orbit a distance $h$ above the surface of the Earth with speed $v_0$. It suffers a head-on collision with some debris which reduces its speed to $kv_0$, where $k$ is a constant in the range $0<k<1$, but does not change its direction. Calculate the eccentricity of the new orbit.We assume that the satellite is being acted on by an attractive inverse square force.
Now before the collision we have
$$\frac{d^2u}{d\theta^2}+u=\frac{\gamma}{L^2}\implies\frac{1}{R} = \frac{\gamma}{R^2 v_0^2} \implies \gamma = Rv_0^2$$
After the collision, $\gamma$ is unchanged, but the angular momentum is.
We shall denote that by $L'$. how do I progress from here? Because I don't know how the magnitude of the angular component of the velocity has changed. What is L'?
Would it be correct to say after the collision $L' = kRv_0$? I feel this is not correct.
I am unsure how to use inline latex on here.
Now before the collision we have
$$\frac{d^2u}{d\theta^2}+u=\frac{\gamma}{L^2}\implies\frac{1}{R} = \frac{\gamma}{R^2 v_0^2} \implies \gamma = Rv_0^2$$
After the collision, $\gamma$ is unchanged, but the angular momentum is.
We shall denote that by $L'$. how do I progress from here? Because I don't know how the magnitude of the angular component of the velocity has changed. What is L'?
Would it be correct to say after the collision $L' = kRv_0$? I feel this is not correct.
I am unsure how to use inline latex on here.