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

1)

Tiffany has determined that the minimum safe pressure is 11 pounds per square inch.

Therefore, we need to substitute, P=11 psi, in the equation, , and find the corresponding value of h.

i.e.,

i.e.,

i.e.,

i.e.,

i.e., miles

i.e., feet

i.e., feet

Therefore the rocket will explode at the altitude of, h=15310 feet.

2)

The height (in feet) of a rocket launched at an angle degrees with the horizontal and an initial velocity, in feet/second, t seconds after launch is given by the equation, .

Here, it is given that, , and,

The maximum height reached by the rocket, will be given by the value of t obtained, when dh/dt=0

i.e., -32t+1648sin(33)=0

i.e., t=(1648sin(33))/32

i.e., t=28.0489103 second

The rocket will reach the maximum height, h(t=28.0489), at t=28.0489 seconds.

h(t=28.0489)=-16(28.0489)^2+1648(28.0489)sin(33)

i.e., h(t=28.0489)=12587.86191 feet

We need to convert the value of h in feet to its value in miles, before we can use it in the equation,  P(h)= 14.7e^(−h/10)

i.e., h=(12587.86191)/5280 miles

i.e., h=2.384064756 miles

The minimum atmospheric pressure exerted on the rocket during its flight, is given by,

i.e., P=11.58187085 psi

i.e., P=11.5 psi, which is greater than the safe limit, of 11 psi. Tiffany has determined that the minimum safe pressure is 11 pounds per square inch.

So we can conclude that under these conditions, the rocket will explode during its flight.

3)

The height (in feet) of a rocket launched at an angle degrees with the horizontal and an initial velocity, in feet/second, t seconds after launch is given by the equation, .

Here, it is given that, , and,

The maximum height reached by the rocket, will be given by the value of t obtained, when dh/dt=0

i.e., -32t+1908sin(32)=0

i.e., t=(1908sin(32))/32

i.e., t=31.59643613 second

The rocket will reach the maximum height, h(t=31.59643613), at t=31.5964 seconds.

h(t=31.5964)=-16(31.5964)^2+1908(31.5964)sin(32)

i.e., h(t=31.5964)=15973.35642 feet

We need to convert the value of h in feet to its value in miles, before we can use it in the equation,  P(h)= 14.7e^(−h/10)

i.e., h=(15973.35642)/5280 miles

i.e., h=3.025256898 miles

The minimum atmospheric pressure exerted on the rocket during its flight, is given by,

i.e., P=10.86255772 psi

i.e., P=10.8 psi, which is lesser than the safe limit, of 11 psi. Tiffany has determined that the minimum safe pressure is 11 pounds per square inch.

So we can conclude that under these conditions, the rocket will not 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.

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.

We have to substitute, P=9 psi, in the equation, P(h)= 14.7e^(−h/10), and solve for h.

i.e., 9=14.7e^(−h/10)

i.e., e^(h/10)=(14.7)/9

i.e.,

i.e.,

i.e.,

i.e., h= 4.906229164 miles

We need to convert the h in miles, to h in feet.

i.e., h= (4.906229164)(5280) feet

i.e., h= 25904.88999 feet

i.e., h=25905 feet

So, the maximum height my rocket can achieve without exploding in flight, is h=25905 feet.