A lunar lander in powered descent at 100 m above the surface of the moon slows down from a velocity of 20 m/s to 0.5 m/s at constant acceleration. Assume that the lander has a mass of 20,000 kg and that burning the fuel does not change the mass. Answer the following questions.
What is the acceleration of the lander?
How long will it take the lander to change velocity from 20 m/s to 0.5 m/s?
What is the thrust (T) of the engines?
To find the acceleration of the lander, we can use the equation of motion:
v^2 = u^2 + 2as
where:
v = final velocity (0.5 m/s)
u = initial velocity (20 m/s)
a = acceleration (unknown to us)
s = displacement (100 m)
Rearranging the equation, we get:
a = (v^2 - u^2) / (2s)
= (0.5^2 - 20^2) / (2 * 100)
Calculating the acceleration:
a = (-399.75) / 200
= -1.9975 m/s^2
So, the acceleration of the lander is approximately -1.9975 m/s^2.
To find the time taken to change velocity from 20 m/s to 0.5 m/s, we can use another equation of motion:
v = u + at
where:
v = final velocity (0.5 m/s)
u = initial velocity (20 m/s)
a = acceleration (-1.9975 m/s^2)
t = time (unknown to us)
Rearranging the equation, we get:
t = (v - u) / a
= (0.5 - 20) / (-1.9975)
Calculating the time:
t = (-19.5) / (-1.9975)
≈ 9.7704 seconds
So, it will take approximately 9.7704 seconds for the lander to change velocity from 20 m/s to 0.5 m/s.
To find the thrust (T) of the engines, we can use Newton's second law of motion:
F = ma
where:
F = force (unknown to us)
m = mass of the lander (20,000 kg)
a = acceleration (-1.9975 m/s^2)
Since we know that force is equal to thrust, we have:
T = ma
= (20,000 kg) * (-1.9975 m/s^2)
Calculating the thrust:
T = -39,950 N
So, the thrust of the engines is approximately -39,950 N. The negative sign indicates that the thrust is in the opposite direction of the lander's motion, which is required to slow down its descent.