(Related Rates)A spacecraft is in free fall toward the surface of the moon at a
ID: 2867954 • Letter: #
Question
(Related Rates)A spacecraft is in free fall toward the surface of the moon at a speed of 1000 miles per hour. Its rctrorockets, when fired, produce a net deceleration of 20,000 mi/hr2. Ignore the effect of the moons gravity.What is the initial condition on the velocity, if the rctrorockets are fired at time t 0 (Be careful withdirection.)Find an equation for the velocity v(t).At what height above the surface should the rectrorockets be fired to ensure a soft touchdown (v = 0) at impact Is it reasonable to ignore the moon's gravity in this situationExplanation / Answer
a)
As can be seen, initially, the velocity is 1000 miles per hour
Lets assume that movement toward the moon is negative movement.
So, initial condition on velocity is : v(0) = -1000 ----> ANSWER to a
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b) a = 20000 mi/hr^2
v(t) = at + v0
v(t) = 20000t - 1000
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c)
x(t) = (1/2)*at^2 + v0*t + x0
Soft touchdown occurs when v(t) = 0
20000t - 1000 = 0
t = 1/20 hour
Now, we find x0
x0 = x(t) - 1/2*at^2 - v0*t
x0 = 0 - 0.5*20000*(1/20)^2 - (-1000)*(1/20)
x0 = 25
So, we must fire the retrorockets 25 miles above the surface to ensure smooth touchdown
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d)
No, it is not reasonable because the moon, being of gigantic mass in comparison to the spacecraft will exert a non-negligible gravitational pull on the spacecraft which will pull the spacecraft in much faster than without any pull.
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