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?
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To find the acceleration of the lander, we can use the kinematic equation:
v^2 = u^2 + 2as
Where:
- v is the final velocity
- u is the initial velocity
- a is the acceleration
- s is the displacement
In this case, the initial velocity (u) is 20 m/s, the final velocity (v) is 0.5 m/s, and the displacement (s) is 100 m. Since we are looking to find the acceleration (a), we rearrange the equation:
a = (v^2 - u^2) / 2s
Substituting the given values:
a = (0.5^2 - 20^2) / (2 * 100)
a = (-399.75) / 200
a = -1.999375 m/s^2
Note that the negative sign indicates that the lander is decelerating.
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 equation:
v = u + at
Where:
- u is the initial velocity
- a is the acceleration
- t is the time taken
In this case, the initial velocity (u) is 20 m/s, the final velocity (v) is 0.5 m/s, and the acceleration (a) is -1.999375 m/s^2. Rearranging the equation to solve for time:
t = (v - u) / a
Substituting the given values:
t = (0.5 - 20) / (-1.999375)
t = (-19.5) / (-1.999375)
t ≈ 9.7526 seconds
Therefore, it will take approximately 9.7526 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 is the force (thrust in this case)
- m is the mass of the lander
- a is the acceleration
In this case, the mass of the lander (m) is 20,000 kg, and the acceleration (a) is -1.999375 m/s^2 (negative because of deceleration). Rearranging the equation to solve for thrust:
T = ma
Substituting the given values:
T = 20,000 * (-1.999375)
T ≈ -39,987.5 N
Therefore, the thrust (T) of the engines is approximately -39,987.5 Newtons.