Estimate the initial velocity the sub-orbital rochket booster needs to have in o

Question

Estimate the initial velocity the sub-orbital rochket booster needs to have in order to gain the desired altitude.

A government missile launch facility is conducting test firings of a new sub-orbital rocket booster. The booster is programmed to follow a parabolic flight path given by the equation: f(x) = -16/v_0^2 x^2 + x On the first firing of the day, with v_0 set at 1000 feet per second, the booster attains an altitude that is 6000 feet less than desired. Find the value of v_0 that the launch officer should use to make sure the booster reaches the desired altitude on the next shot. How far down range did the booster impact on the first firing? What was the maximum height attained by the booster on the first firing? Value of v_0 so the booster reaches the desired altitude:  How high did the booster go on the second firing? How far down range did the booster impact on the second firing? What are the domain and range of f(x) for this problem? Imagine you are a tutor and have to explain this problem to another student. Write a one-paragraph (3 sentences or more) explanation of how to solve this problem

Explanation / Answer

f(x) = -16x^2/1000^2 +x

for maximum height, df/dx = 0

-32x/1000^2 +1 =0

x = 1000^2/32 =31250

f(x) = -16*31250^2/1000^2 + 31250 = 15625 ft

Required altitude = 15625+6000 =21625 ft

maxima = v^2/32

f(x) = 21625 =  -16* ( v^2/32)^2 /v^2 +v^2/32 = - v^2/64 +v^2/32 = v^2/64

v/8 = sqrt21625

v = 1176 ft/s

Required initial velocity is 1176 m/s

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