Rockets Away: Quadratics and Newton’s Third Law of Motion

Background:Sir Isaac Newton (1643 – 1727) was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian and one of the most influential men in human history. His Philosophiæ Naturalis Principia Mathematica, published in 1687, is considered to be the most influential book in the history of science, laying the groundwork for most of classical mechanics. In this work, Newton described universal gravitation and the three laws of motion which dominated the scientific view of the physical universe for the next three centuries.1

Newton's laws of motion can be qualitatively summarized by the statement that changes in motion of objects are caused by forces acting on them. Thus, any object experiencing acceleration is necessarily under the influence of a force.2

The launching of a rocket can be explained by Newton’s third law of motion: for every action there is an equal and opposite reaction. On a very basic level, a rocket is a pressurized gas chamber. As gas escapes through a small opening at one end, the rocket must be propelled in the opposite direction as stated by Newton’s third law. The force that propels the rocket is called thrust, as noted in the diagram3 below:

The flight of a model rocket follows a parabolic path and exhibits projectile motion. A projectile is an object upon which the only force acting is gravity. Once the rocket burns its fuel, inertia carries the rocket upward until the pull of gravity forces it back to earth. The rocket’s projectile motion can be described using a simple quadratic function.

Investigation:

Part 1: Constructing a RocketTo better understand the relationship between quadratics and Newton’s Third Law of Motion, you will construct and fly a water bottle rocket. The rocket must meet the following guidelines:

Body must be made from a 2 liter soda bottle Fins (four works well) can be constructed from heavy cardboard, attached with duct tape,

and should not extend beyond the neck of the bottle Nose cone should have a small mass (a small bag of sand inside the cone works well) The body must remain water-tight; do not cut or hot-glue the bottle Remember: the bottom of the bottle is the top of the rocket; do not cover the bottle opening

Part 2: Flying the RocketYou will have one opportunity to launch your rocket. For your flight, you will need to record the following data:

Live Flight DataVolume of Water (mL) Pressure (psi) Time Up (sec)

(to nearest tenth)Time Down (sec) (to nearest tenth)

Your flight will also be recorded on video. When you are filming, be sure to capture the entire flight path so that you can extract time data from the video.

Part 3: Describing Flight Mathematically—Quadratic Functions

Mathematicians have developed a formula for describing projectile motion. This quadratic function can be expressed as:

t = time in secondsh(t) = height in meters after a certain amount of time: example h(2) means height after 2 seconds.vo = starting vertical velocity or vertical velocity in m/s when time = 0 secondsho = starting height in meters or height when time = 0 secondsacceleration due to gravity is -9.8 m/s2

If we assume that the initial height and velocity are zero, the maximum height of the rocket can be calculated using Time Up and Time Down. To calculate the maximum height of your rocket based on Time Up and Time Down, you will need to determine the vertex of the parabola. Explain why.

1. Write the expression for finding the x-coordinate of the vertex of a parabola.

2. Substitute your known values for variables “x” and “a” (“x” = time up and “a” can be determined form the equation at the top) to determine variable “b” in #1.

3. Using the standard form for a quadratic substitute appropriate values for the variables a, b and c. Then rewrite your equation using h(t) for y, and t for x.

4. Find the maximum height (the x-coordinate of the vertex) using Time Up.

5. Repeat steps #1-3 above to find the maximum height (the x-coordinate of the vertex) using Time Down.

6. In theory, Time Up and Time Down should be equal. If these values were not equal for your rocket, explain what could cause them to differ. Which time value (Up or Down) would produce a more accurate height calculation?

7. Does it make sense to adjust your equation at this point? If yes, make proper changes. Otherwise, choose the equation you think produces the more accurate height and enter it into your graphing calculator. Adjust the windows and scale so your parabola fits your screen.

8. Below, write the values you chose for x-min, x-max, x-scl, y-min, y-max, y-scl.

9. On grid paper, draw an accurate graph of your equation, labeling axes and using appropriate scales.

10. Using either your equation or the table from your calculator, at what point(s) in time does:

a. h = 5 m

b. h = 15 m

c. h = 25 m

Part 4: Describing the Flight Mathematically—Film Data

Import your flight video into the computer. The playhead in the editing software will mark time to 1/29 of a second (each second of motion on film is composed of 29 still frames). Move the playhead to determine the (1) Time Up and (2) Time Down for your rocket.

Film Flight DataTime Up (sec) Time Down (sec)

1. Recalculate the maximum height of your rocket using either Time Up or Time Down (choose the more accurate of the two) from your film flight data:

2. What is the velocity of the rocket on impact (based on the film data)? Hint: consider acceleration due to gravity and time

Part 5: Modeling the Flight—Computer Simulation

There is no substitute for real-world data, but simulations can be very useful (and accurate) tools. Water Rocket Fun from Seeds Software can model the flight characteristics of a water bottle rocket. Open this program and on the Set Up tab, enter the data for your rocket:

From the Flight Tab, “launch” the rocket and record the flight data below:

Time (s) Max Height (m) Max Velocity (m/s)

3. How do these values compare to the time, height, and velocity values you calculated using your field data? Film data? Do you think they are more or less accurate? Why?

From the Graph Tab, plot the Height vs. Time for the flight (up and down):

4. How does this compare to the graph you made for the field data? At what points in time does:a. h = 5m

b. h = 15m

c. h = 25m

Conclusion:

The investigation you conducted utilized Newton’s Third Law of Motion to illustrate a quadratic function. By measuring data in the field, on film, and via a simulation, you have three comparisons for your rocket launch.

1. In your study of second degree equations, you discovered that a quadratic equation may have one solution, two solutions or no solutions. Explain how that fact relates to the Rocket Project.

2. Discuss areas where your data fit and areas where it did not fit in the three comparisons.