Let be the speed of a body moving in a circular orbit of radius around a planet of mass

PHYSICS · Multi-part · A Level · Physics · Fields and their consequences · Fields and their consequences

Problem

Let

v_{\text{c}}

be the speed of a body moving in a circular orbit of radius

r

around a planet of mass

M

, and

v_{\text{esc}}

be the minimum speed required for a body to leave the planet’s gravitational influence when launched from the same radius

r

. (i) Derive the relation

v_{\text{esc}} = \sqrt{2}\,v_{\text{c}}

. (ii) A spacecraft is in a circular orbit

400\,\text{km}

above Earth’s surface. Take Earth’s mass

M_{\oplus}=6.0\times10^{24}\,\text{kg}

and radius

R_{\oplus}=6.4\times10^{6}\,\text{m}

. Calculate (a) the orbital speed

v_{\text{c}}

at this altitude, and (b) the corresponding escape velocity

v_{\text{esc}}

from that altitude. (iii) Determine the additional kinetic energy per kilogram that must be supplied to the spacecraft so that it can move from the circular orbit into an escape trajectory from Earth.

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