ACTUAL ROCKET SCIENCE
NAME: ______________________________ Period: ____
An amateur rocket flight has two stages: a boost stage and a free-flight stage. When a rocket launches, it accelerates until it uses all of its fuel, then gravity takes over and the rocket returns to Earth. In the following examples, each rocket is equipped with a janky GPS / accelerometer that produces incomplete data.
ROCKET #1 – DÜMD2FAIL For this first example, assume DÜMD2FAIL’s acceleration does not change during its boost stage A) During boost stage, DÜMD2FAIL’s GPS only records two velocities. At 0.321 seconds into the flight, the rocket is traveling at 10.8 m/s and at 0.960 seconds into the flight, the rocket is traveling at 32.3 m/s. Determine the acceleration of DÜMD2FAIL during its boost stage suing the slope formula.
B) DÜMD2FAIL’s boost stage lasted 2.23 seconds. Assuming the acceleration calculated in Part A was constant, what was the rocket’s velocity when its fuel ran out? SHOW YOUR WORK!! v0: ____________ a: ____________ vf: ____________
C) After the boost stage, DÜMD2FAIL’s height was 83.660m. Use the velocity from Part B to write a vertical motion equation to describe the height of the rocket after the boost stage ends. h(t) = D) In Geogebra, type the following lines of code in the input bar. Make each graph a different color. if[x>0&&x<2.23,16.823*x^2]
if[x>2.23&&x<22,-4.9*(x-2.23)^2+75.031*(x-2.23)+83.66]
E) What did you do?!? Explain what the lines of code above do: i) What does “x>2.23&&x<22” mean? ii) The generic code for an if statement is “if[ , , ]”. Explain how this code was used in this case:
iii) Why is there a “(x – 2.23)” in the second equation? Where have you seen that before? What is the significance of the -4.9 and the 16.823 in each equation?
F) Use the max[,,] command to find at what time the maximum height of the rocket is achieved:
G) Use the root[,,] command to find when the rocket hits the ground:
H) During the boost stage, what is the rocket’s acceleration? I) After the boost stage, what is the rocket’s acceleration? Careful! J) Draw an acceleration vs time graph to represent the rocket’s flight. Important times have been plotted.
K) After the boost stage, describe what will happen to the rocket’s velocity. In this situation, is it OK to have a negative velocity? Is there a point in the flight when DÜMD2FAIL’s velocity is zero? (Explain)
L) Taking into account all information, draw an accurate velocity vs time graph for DÜMD2FAIL.
ROCKET #2 – SPЦTПIК IѴ In reality, the acceleration of a rocket is not constant as modeled by DÜMD2FAIL. If you consider a rocket is just a tube full of fuel, then the mass of the rocket will decrease significantly as the fuel burns. Meanwhile, the power produced from the engine remains fairly constant… meaning the same force is now powering an object that is getting lighter, resulting in increased acceleration. SPЦTПIК IѴ is a considerably larger and more powerful rocket than DÜMD2FAIL. The engine produces 400N of thrust force during a boost stage 6.21 seconds long. At 2.3 meters tall, the fully fueled rocket has a mass of 25kg and a mass of 7kg when empty. Answer the following questions about SPЦTПIК IѴ: A) How much mass does SPЦTПIК IѴ lose during its boost stage?
B) Write a linear function that describes the mass of the rocket vs. time:
C) Write a well-known formula that relates the force, mass, and acceleration of an object. (Google?) Solve this equation for acceleration.
D) Substitute your expression for the mass of SPЦTПIК IѴ into your equation from Part C. This equation will now describe the acceleration of the rocket during the boost stage. To complete your acceleration equation, substitute in the value for the thrust force of the rocket (above).
E) Use Geogebra to help you make an accurate plot of the acceleration function during the boost stage:
F) Zoom out and notice your acceleration function has an asymptote. Give an equation for the asymptote. Explain why it occurs and why, in the context of this problem, it does not matter.
G) Based on what you know about acceleration, velocity, and position vs time graphs, explain how you will find the velocity of SPЦTПIК IѴ after the boost stage:
H) Explain why finding the area under the acceleration curve is significantly more difficult than with DÜMD2FAIL:
I) Explain how you will produce an accurate estimate of the area under the curve:
J) Use at least 4 polygons to estimate the area under the acceleration curve. SHOW YOUR WORK!! Attach a label to your answer.
K) What does your answer to J represent? (be specific)
L) Is your estimate an over-estimate or under-estimate? Explain how you know.
Geogebra has built in commands to find the area under a curve M) With your acceleration function from Parts D & E plotted in Geogebra, use the Integral[,< start value>, ] command in Geogebra to find the exact area under the acceleration curve for the boost stage.
N) What is the velocity of the rocket after the boost stage?
O) After the boost stage, SPЦTПIК IѴ’s GPS recorded an altitude of 432m. Using this information and your answer to Part N, write a vertical motion equation that models the rocket’s free flight stage.
P) Graph your equation from Par O and find the max height of the rocket and when it hits the ground.
max height: _____________
time of touchdown: ____________
Q) Make a rough sketch of the velocity and acceleration functions, including the boost stage:
ROCKET #3 – iXPLODE Now consider a rocket where the force produced by the engine is not constant. As fuel is burned in the chamber, an air pocket develops, which allows more and more hot gas to compress before it is expelled out the nozzle. As a result, the thrust produced by the engine decreases, slightly at first, then much quicker as the air pocket increases in size and even more quickly as the remaining fuel is used (see below).
A) Use a function transformation to write an equation that closely mimics the above relationship. Your equation does not have to be absolutely perfect, but it should follow the general trend, shape, and magnitude of the above relationship. Make sure your model shuts off the engine at 4.25 seconds. Write your force vs. time equation here:
B) iXPLODE has a mass of 20kg when fully fueled and contains 16kg of fuel. Write a linear equation that describes the mass of the object during the boost stage.
Write your mass vs. time equation here:
C) Use a similar method to Parts C and D of SPЦTПIК IѴ to write an acceleration function for iXPLODE.
Write your acceleration function here:
D) Does your acceleration function have any domain restrictions? If so, list them and explain whether or not they will cause a problem within the context of this problem.
E) Graph your acceleration function using a table of values:
F) Use the Integral[,< start value>, ] to find the final velocity after the boost stage. velocity after boost stage: ___________________ G) The GPS on iXPLODE recorded a height of 170m after the boost stage. Write a vertical motion equation to model the height of the rocket during the free flight stage: h(t) = H) Use your equation in Part H to find the max height of the rocket and when it impacts the ground:
max height: _____________
time of touchdown: ____________
I) Write 2 lines of code for Geogebra that will plot the entire position vs. time graph of iXPLODE (See Part D of DÜMD2FAIL for reference) Sketch your graph below:
