Past exam of the physics 1 paper of the University of Oxford / 2013 resit / 10 / b

Solution

For a tangential launch, conserved specific angular momentum is

j = R v

. At the outer turning point the radial velocity vanishes but the tangential speed is

j / r_{0}

. Thus

\frac{1}{2} v^{2} - \frac{μ}{R} = \frac{R ^{2} v ^{2}}{2 r _{0}^{2}} - \frac{μ}{r _{0}} .

v = \frac{v _{0}}{1 - R ^{2} / r _{0}^{2}} = v_{0} [1 + \frac{R ^{2}}{2 r _{0}^{2}} + O (\frac{R ^{4}}{r _{0}^{4}})] .

The expression printed in the source is the leading correction for

r_{0} ≫ R

, rather than an exact identity. Equivalently,

v^{2} = 2 μ r_{0} / [R (r_{0} + R)]

, so the launch speed exceeds the circular value and the surface is the periapsis of the nonintersecting ideal orbit.

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