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 first equation of motion:
v² = u² + 2as
where:
v = final velocity (0.5 m/s)
u = initial velocity (20 m/s)
a = acceleration of the lander
s = distance traveled (100 m)
Rearranging the equation to solve for acceleration (a), we get:
a = (v² - u²) / (2s)
Plugging in the values we have:
a = (0.5² - 20²) / (2 * 100)
Calculating this expression gives us:
a = (-400 + 0.25) / 200
a ≈ -1.9975 m/s²
Thus, the acceleration of the lander is approximately -1.9975 m/s².
To find the time it takes for the lander to change velocity from 20 m/s to 0.5 m/s, we can use the second equation of motion:
v = u + at
where:
v = final velocity (0.5 m/s)
u = initial velocity (20 m/s)
a = acceleration of the lander (-1.9975 m/s²)
t = time taken
Rearranging the equation to solve for time (t), we get:
t = (v - u) / a
Plugging in the values we have:
t = (0.5 - 20) / -1.9975
Calculating this expression gives us:
t = -19.5 / -1.9975
t ≈ 9.76 seconds
Thus, it will take approximately 9.76 seconds for the lander to change its 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 = net force acting on the lander
m = mass of the lander (20,000 kg)
a = acceleration of the lander (-1.9975 m/s²)
Since the lander is in a powered descent, the net force is equal to the force of gravity, which is given by:
F = mg
where:
g = acceleration due to gravity on the moon (approximately 1.6 m/s²)
Plugging in the values we have:
F = 20,000 kg * 1.6 m/s²
F = 32,000 N
Since force (F) is equal to mass (m) multiplied by acceleration (a), we can write:
mg = ma
Replacing the values:
20,000 kg * 1.6 m/s² = 20,000 kg * -1.9975 m/s² + T
Calculating this expression gives us:
32,000 N = -39,950 N + T
Rearranging the equation to solve for thrust (T), we get:
T = 32,000 N + 39,950 N
T ≈ 71,950 N
Thus, the thrust of the engines is approximately 71,950 N.