Question

You have entered a model rocket contest with your friend, Tiffany. You have been working on a pressurized rocket filled with nitrous oxide. Tiffany has determined the minimum atmospheric pressure at which the rocket fuel is stable. Based not hat value, and the equations given below, your task is to determine the optimum launch angle and initial velocity to maximize flight time. The goal is to re-use your rocket capsule, so you really want to avoid a fuel explosion.

The atmospheric pressure varies with elevation according to the equation: P(h)= 14.7e−h/10. where p is the pressure in psi and h, is an elevation in miles above sea levels. The height (in feet) of a rocket launched at an angle α degrees with the horizontal and an initial velocity, vo in feet/second, t seconds after launch is given by the equation h(t)=-16t^2+vo*t*sin(α).

1) If Tiffany has determined that the minimum safe pressure is 11 pounds per square inch, at what altitude will the rocket explode? Report your result in feet. Round to the nearest foot.

2) If the angle of launch is33o, with an initial velocity of 1,648 what is the minimum atmospheric pressure exerted on the rocket during its flight? Report your answer to one decimal place. Under these conditions, will the rocket explode during its flight?

3) If the angle of launch is32o, with an initial velocity of 1,908 what is the minimum atmospheric pressure exerted on the rocket during its flight? Report your answer to one decimal place. Under these conditions, will the rocket explode during its flight?

4) Tiffany has revisited her calculation and has now concluded that the minimum safe pressure for the fuel is 9 psi. What is the maximum height your rocket can achieve without exploding in flight? Report your answer in feet to the nearest foot.

5) Tiffany has (once again) checked her calculations, and you have verified with her that the safe pressure for your fuel is 9 and the fuel capsule holds enough fuel to produce an initial velocity of 2,169 feet per second. What launch angle will you use so that your rocket achieves the maximum safe altitude? Round your answer to the nearest tenth of a degree.

Answer 1

For the given rocket problem
pressure variation with elevation is P(h) = 14.7*e^(-h/10), P is psi and h in miles
and
h(t) = -16t^2 + vo*t*sin(alpha)
  
a. h = -16t^2 + vo*sin(alpha)*t
h'= -32t + vo*sin(alpha)
h"= -32
as we can see, the only acceleration acting is due to gravity

minimum safe pressure is 11 psi
11 = 14.7*e^(-h/10)
so the rocket explodes at a height of 2.89952220986 miles = 15309.47726806 ft
  
b. alpha = 33 deg
vo = 1648 ft/s
then
h = -16t^2 + 1648*sin(33)*t
16t^2 - 897.5651297 + h = 0

at max height, h'= 0
hence,
t = vo*sin(alpha)/32 = 28.04891030 s
hence,
h = 12587.8619 ft = 2.38406 miles
so minimum pressure is 14.7*e^(-2.384/10) = 11.58187085 psi
the rocket will not explode

c. alpha = 32 deg,
vo = 1908 fps
16t^2 - 1908*sin(32)*t + h = 0
t = 1908*sin(32)/32 = 31.59643612990 s
h = 15973.356417 ft = 3.0252568 mi
minimum pressure = 14.7*e^(-3.025/10) = 10.86283 psi < 11 psi
hence the rocektexplodes
  
d. minimum pressure now is 9 psi
now maximum height is h
9/14.7 = e^(-h/10)
h = 4.9062291644 miles = 25904.889988 ft
  
e. safe pressure is 9 psi,
so max altitude is 25904.8899 ft
t = 2169*sin(a)/32
25904.88998847928 = 16t^2
t = 40.237490283 s
hecne, a = 36.41 deg