A lunar-mapping probe of negligible mass orbits Earth on the highly eccentric ellipse…

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

Problem

A lunar-mapping probe of negligible mass orbits Earth on the highly eccentric ellipse shown below. The probe’s apogee distance is

r_a = 4.8 \times 10^{8}\,\text{m}

(just beyond the Moon’s orbit) and its perigee distance is

r_p = 1.0 \times 10^{7}\,\text{m}

. At apogee the probe’s speed is measured to be

v_a = 180\,\text{m/s}

. Assume Earth’s mass is

M_E = 6.0 \times 10^{24}\,\text{kg}

. (a) Derive an expression, in symbolic form, for the probe’s speed

v_p

at perigee in terms of

v_a

r_a

, and

r_p

only. (b) Calculate the numerical value of

v_p

. Express your answer to two significant figures. (c) Determine the semi-major axis

a

of the orbit. (d) Compute the total mechanical energy

E

of the Earth–probe system and state whether the orbit would remain bound if

v_a

were increased by 20 %. Justify your answer.