Question

The Apollo Lunar Module was used to make the transition from the spacecraft to the Moon's surface and back. Consider a similar module for landing on the surface of Mars. Use conservation of mechanical energy to answer these questions.

(a) As the lander is descending, if the pilot decides to shut down the engine when the lander is at a height of 2.8 m, (this may not be a safe height to shut down the engine) and the velocity of the lander (relative to the surface of the planet) is 1.3 m/swhat will be velocity of the lander at impact? Note: g on the surface of Mars is about 0.4 times that on the surface of the Earth.


(b) In the case of the lunar module an impact velocity of 3.0 m/s or less was essential for a safe landing. Assuming this to be the case for the Mars lander as well, at what maximum height could the pilot shut down the engines to ensure a safe landing. Assume the velocity v₀ at the time the engine is shut down is 1.3 m/s.

Answer 1

a)

h = height = 2.8 m

v_o = initial speed = 1.3 m/s

v_f = final speed = ?

g = acceleration = 0.4 x 9.8 = 3.92 m/s²

m = mass

using conservation of energy

kinetic energy at the bottom = kinetic energy at Top + potential energy at top

(0.5) m V_f² = (0.5) m V_o² + mgh

V_f² = V_o² + 2gh

V_f² = (1.3)² + 2 (3.92 x 2.8)

V_f = 4.86 m/s

b)

h' = height at which the engine is is shut down

v_o = initial speed = 1.3 m/s

v_f = final speed = 3 m/s

g = acceleration = 0.4 x 9.8 = 3.92 m/s²

m = mass

using conservation of energy

kinetic energy at the bottom = kinetic energy at Top + potential energy at top

(0.5) m V_f² = (0.5) m V_o² + mgh'

V_f² = V_o² + 2gh'

3² = 1.3² + 2 (3.92) h'

h' = 0.93 m