Paper 4, Section II, 10C

Dynamics and Relativity | Part IA, 2021

(a) A mass mm is acted upon by a central force

F=kmr3r\mathbf{F}=-\frac{k m}{r^{3}} \mathbf{r}

where kk is a positive constant and r\mathbf{r} is the displacement of the mass from the origin. Show that the angular momentum and energy of the mass are conserved.

(b) Working in plane polar coordinates (r,θ)(r, \theta), or otherwise, show that the distance r=rr=|\mathbf{r}| between the mass and the origin obeys the following differential equation


where hh is the angular momentum per unit mass.

(c) A satellite is initially in a circular orbit of radius r1r_{1} and experiences the force described above. At θ=0\theta=0 and time t1t_{1}, the satellite emits a short rocket burst putting it on an elliptical orbit with its closest distance to the centre r1r_{1} and farthest distance r2r_{2}. When θ=π\theta=\pi and the time is t2t_{2}, the satellite reaches the farthest distance and a second short rocket burst puts the rocket on a circular orbit of radius r2r_{2}. (See figure.) [Assume that the duration of the rocket bursts is negligible.]

(i) Show that the satellite's angular momentum per unit mass while in the elliptical orbit is

h=Ckr1r2r1+r2h=\sqrt{\frac{C k r_{1} r_{2}}{r_{1}+r_{2}}}

where CC is a number you should determine.

(ii) What is the change in speed as a result of the rocket burst at time t1t_{1} ? And what is the change in speed at t2t_{2} ?

(iii) Given that the elliptical orbit can be described by

r=h2k(1+ecosθ)r=\frac{h^{2}}{k(1+e \cos \theta)}

where ee is the eccentricity of the orbit, find t2t1t_{2}-t_{1} in terms of r1,r2r_{1}, r_{2}, and kk. [Hint: The area of an ellipse is equal to πab\pi a b, where aa and b are its semi-major and semi-minor axes; these are related to the eccentricity by e=1b2a2.]\left.e=\sqrt{1-\frac{b^{2}}{a^{2}}} .\right]

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