C-2Improving the Range Equation
1Understanding the quadratic equations
a. Write the new equation.
b. Quadratic equations have the form ax2 + bx + c = 0, where a, b, and c are constants and x is a variable. Your equation is a quadratic equation for t, the time of flight for the marble. Take your equation and group the terms so that they look like a quadratic equation for t. Write our equation out in quadratic form.
c. Solve the equation for t using the quadratic formula. One of the solutions will be positive, corresponding to the marble hitting the ground in front of the marble launcher. The other solution will be negative. What does this solution correspond to?
d. Use the positive solution and derive an equation for the range (x) by substituting your results for t into x=vtcos. Your resulting equation should only depend on the initial velocity v and the launch angel .
e. Since you didn’t do the original equation, us the equation
to solve for the Old Theoretical Value.
2Testing the new theory
a. Do you expect that the new and old theories will differ most at small or large angles?
b. Will the differences be greatest at small or large speeds.
c. Use the answers to these questions to determine the data you think will best demonstrate the differences between the new theory and the old theory. Use the table below to record your data and calculations.
Launch Angle (degrees) / Spring Setting / Time A(sec) / Initial Velocity
(m/s) / Measured Range (m) / Old
Theoretical Range (m) / New
Theoretical Range (m)
10o / 1
20o / 1
40o / 1
60o / 1
10o / 2
20o / 2
40o / 2
60o / 2
10o / 3
20o / 3
40o / 3
60o / 3
10o / 4
20o / 4
40o / 4
60o / 4
10o / 5
20o / 5
40o / 5
60o / 5
d. Make four graphs, one for each of the different launch settings (notch). The graphs are to be range vs. angle. In each one, use a circle for the actual result, a triangle for the old theory values, and a square for the new theory values. Draw the best possible curve through the new theory values in one color, and a curve through the old theory values in another color. Do not draw a curve for your actual values.
e. Calculate the two sets of absolute and relative error values, using your old theory values as accepted in the first table, and new theory values as accepted in the second table.
Error Table 1 (old theory values)Error Table 2 (new theory values)
Measured Range (m) / Old Theoret. Range (m) / Relative Error (m) / Absolute Error (%) / Measured Range (m) / New Theoret. Range (m) / Relative Error (m) / Absolute Error (%)f. How does your new theory compare to your actual measurements?
g. From your measurements, can you see any other effect that changes the range of the marble and could be included in your theory?
Questions
1The marble launcher is set up with an angle of 65o and a range of 340 cm is measured. What other angle will give the same range at the same initial velocity? (For this problem, you may ignore the initial height of the launcher.
2The marble launcher is set up with an angle of 80o and the timer measures 0.0030 second for the 0.019 m marble to pass through the beam. The marble is launched from a height of 1.1 m off the floor. Calculate the range of the marble measured from the edge of the table, aligned with the front of the marble launcher.
3A stunt motorcyclist claims he will jump over 10 buses lined up side-by-side. The line of buses is 55 m wide. The ramp he uses is at a 45o angle. How fast must he drive when he leaves the ramp if he wants to make the jump?