Question

On Apollo Moon missions, the lunar module would blast off from the Moon's surface and dock with the command module in lunar orbit. After docking, the lunar module would be jettisoned and allowed to crash back onto the lunar surface. Seismometers placed on the Moon's surface by the astronauts would then pick up the resulting seismic waves.

Find the impact speed of the lunar module, given that it is jettisoned from an orbit 100 km above the lunar surface moving with a speed of 1600 m/s .

Answer 1

Given,

h = 100 km ; v = 1600 m/s

from conservation of energy

(KE + PE) = (KE + PE)at crash

1/2 m v^2 - GMm/R = 1/2 m v'^2 - G Mm/Rm

v^2/2 - GM/R = v'^2/2 - GM/Rm

v' = sqrt (v^2 + 2GM(1/Rm - 1/R'))

R = Rm + h ; where Rm is moon's radius ; M = 7.35 x 10^22 kg ; Rm = 1737000 m

R = 1737000 + 100000 = 1837000 m

1600^2/2 - 6.67 x 10^-11 x 7.35 x 10^22/(1837000) = v'^2/2 - 6.67 x 10^-11 x 7.35 x 10^22/(1737000)

v'^2/2 = 1600^2/2 - 6.67 x 10^-11 x 7.35 x 10^22/(1837000) + 6.67 x 10^-11 x 7.35 x 10^22/(1737000) = 1.433 x 10^6

v' = sqrt (2 x 1.433 x 10^6) = 1693 m/s

Hence, v' = 1693 m/s