Phil Draviam Gabe Dagani
October 30, 2003
Unit Title:Evil Knievel
Instructional Goals:
1. Students should understand the relationship between force, mass and acceleration.
2. The students will use trigonometry and algebraic techniques to determine the horizontal and vertical positions of a launched object.
Instructional Objectives:
1. Given a constant force with varying masses, the students will be able to applyNewton's second law to determine how the acceleration of the object is affected.
2. Using the trajectory and position equations, students will predict the distance and height traveled of a launched object within one foot.
3. After measuring the angle of inclination, the students will use trigonometric relations to understand how the angle affects the distance traveled.
4. Using graphing techniques, students will calculate and plot the height and distance at specific times.
Rationale:
Students should know how a trajectory works because it is an essential part of every day physical behavior. Knowing how specific changes in angle, velocity, and gravity influence an object in motion will help the students understand how objects move in real life. Being able to plot graphs and predict trajectories using graphs will better illustrate this behavior as it engages the student.
Content:
Science Standards:
Physical Sciences
21. Demonstrate that motion is a measurable quantity…
22. Demonstrate that any object does not accelerate …
23. Explain the change in motion (acceleration) of an object…
25. Demonstrate the ways in which frictional forces constrain…
Science And Technology
2. Identify a problem or need, propose designs and choose…
Scientific Inquiry
3. Construct, interpret and apply physical and conceptual models….
Math Standards:
Measurement Standard
4. Use scale drawings and right triangle trigonometry to solve…
Geometry And Spatial Sense Standard
1. Define the basic trigonometric ratios in right triangles…
2. Apply proportions and right triangle trigonometric ratios to solve…
Patterns, Functions And Algebra Standard
12. Simplify rational expressions by eliminating common factors…
Learning Activities:
------
1st Day (60 minutes) – Math class only
------
(10 minutes) Introductory video
Show students a short video of various stunt drivers performing dangerous stunts. The purpose of the video is to capture the attention of the students and introduce the activity. Ask the students questions such as:
“How do the drivers in the film know how far they can jump?”
“How would you figure it out?”
“What factors influence the distance the stunt drivers can jump?”
a) Acceleration (In our case, vehicle will be decelerating)
b) Mass of the object (Why don’t they jump with busses? Require large force)
c) Material of the ramp (Frictional force)
d) Wind (Frictional force)
e) Angle of the ramp
f) Heights of the launch/landing ramps
(10 minutes) Project description
Show students the cars, launcher and describe the objectives of the activity. Mention that the students will receive personal folders to document the worksheets and data from the activity. Students will work in teams of three and should compare calculations with other group members for thoroughness. Prizes will be awarded for the top team in each class.Clearly explain how their performance will be evaluated, and to what degree of accuracy their calculations should be performed.
(30 minutes) Newton’s first and second laws
Discuss Newton’s first law and ask students to provide examples of where it is observed.
Within the activity, it is observed as the car requires a force to move from rest. Since the ramp is not frictionless, the car slows down as it travels.
Explain what the variables in F = ma represent and how the acceleration, force and mass are related. Provide examples of this law.
(10 minutes) Measurement of angles
Pass out individual folders. Emphasize that the folders will remain in the classroom for student use.
Divide students into groups of three. Remind students that they are in competition with other teams, and that they should only work and collaborate among team members.
Students measure the mass of the car and the angle of the three various ramp heights. Students are to record their calculations and data in their designated folders, as their folders will be collected for a grade.
------
2nd Day (120 minutes) – Block class for physics and math
------
(40 minutes) – Basic Trigonometry
Explain SOH-CAH-TOA and how it works. Define where the hypotenuse, adjacent and opposite sides are on a triangle. Work through a couple examples on the board while students work at their desk using the calculators.
Pass out the worksheet for students to practice using trigonometry to determine lengths and angles. Students will also gain practice on using calculators to solve trigonometric and inverse-trigonometric functions.
(20 minutes) – Trajectory Equations
Introduce the trajectory equations and what the various components represent. Students should not be concerned with the derivation of the equation, but should understand how it may be applied. Explain what the various variables represent, and that gravity (g) is a constant value. Ask students what values of x and y are desirable and relate the equation to the problem at hand.
Vertical position:
y = viy*t + (0.5)*g*t2
Horizontal position:
x = vix*t
Draw a triangle of the ramp and tell the students that we know the velocity in the angled direction, but how can we determine what viyand vix are. Refer to the previous exercise on trigonometry.
(30 minutes) – Solve for one ramp
As a class, work through the solution for the equations given a ramp of a specified angle. Select one of the ramp angles that was measured yesterday and solve for the horizontal distance. Begin by solving for viyand vixby using the trigonometric relations. Next, plug values into the vertical position equation. Finally, \solve the equation for the horizontal position.
Draw an x-y coordinate plane on the board and show the students where the car is located at the start (x=0) and finish (when y=0). Indicate that these are two coordinate pairs, and that we are interested in what to car does between those two points.
(20 minutes) – Plotting
Have the students solve the equations for five different time intervals between the launch and landing of the car. Have the students create their own x-y plots and have them plot the trajectory of the car from the calculated data points.
------
3rd Day (120 minutes) – Block class for physics and math
------
(60 minutes) – Teamwork
Students work as a team to solve the trajectory equations for the remaining ramps of two different heights. They will determine the horizontal distance traveled by the car in each case. Further, students will calculate the time for each jump, and select 5 time intervals within that to gather data for plots. Students will plot the trajectories for each of the two remaining ramps.
If groups finish early, propose the calculation for a ring to be placed within the jump.
(40 minutes) – Testing
Each team will submit a piece of paper with the following before testing:
a) Which ramp angle they want to use
b) How far the landing ramp should be from the starting ramp
c) Optional – Where the vertical ring should be placed
Students will record in their notebook this information and will
(20 minutes) - Review
Write the results for each team on the blackboard and compare results.
Discuss possible reasons for failure or success.
Example: Inaccurate measurement of the ramp height and length leads to incorrect angles and compounding effects in the calculations.
Example: Inconsistent launch speeds.
Example: Rounding of numbers during calculations.
Materials:
Day 1
Laptop
Projector
5 minute video compilation
Hot wheel racecars
Track
Ramp
Launcher
Booster
Posts to elevate ramp
Individual folders
Worksheet 1
Day 2
Worksheet 2
One sheet of graph paper per student
Worksheet 3
Day 3
Two sheets of graph paper per student
Worksheet 4
Measuring tape
NEED TO DISCUSS
1. Evaluation of the activity
Folders will be collected for a grade
Post activity quiz?
2. Assessment of student learning
Pre and post testing?
3. Further ideas for Newton’s Laws presentation
HANDOUTS
Worksheet 1 – Space for mass of cars
Space to write down the heights of the three ramps
Space to write down the lengths of the three ramps (constant)
Space to write down the base of the three ramps
Space to write down all three angles for each ramp
Worksheet 2 – Practice trig worksheet with approx. 10 problems
Worksheet 3 – Trajectory equations and room for students to write down derivations
Illustration of ramps and the x and y directions
Worksheet 4 – Contains the final calculations
Includes spaces for the x distance traveled for each ramp
Will also include space for the experimental results
Topics:
Physics:
1. Force = Mass * Acceleration
2. Object in motion remains in motion (discuss frictional forces)
3. Have students measure the angles of the different ramps?
Math:
1. Trigonometry for determining x and y velocities based on the angle of the ramp.
2. Solve position equation in the y-direction to determine when the object will return to the initial y-position. (Solve for y=0)
3. Use the calculated flight time of the object to determine how far it will travel in the x direction.
4. Calculate the x and y position of the object at 5 different times before the object returns to the ground state, and graph the position of the object at each calculated time.
5. Repeat plots for all three ramps of varying angles.
Student Challenges:
1. Which angled ramp will produce the greatest distance traveled in the x direction?
2. What combination of cars/trucks can be placed between the two ramps?
3. Why does the velocity at the top of the ramp decrease with ramp steepness, when the initial velocity remains the same?
4. Will a car or motorcycle travel further if propelled with the same initial force?
5. Reflect on some possible reasons for error in the final calculations.