1. (7 points, suggested time 13 minutes)
A stunt cyclist builds a ramp that will allow the cyclist to coast down the ramp and jump over several parkedcars, as shown above. To test the ramp, the cyclist starts from rest at the top of the ramp, then leaves the ramp,jumps over six cars, and lands on a second ramp.
H0 is the vertical distance between the top of the first ramp and the launch point.
θ0 is the angle of the ramp at the launch point from the horizontal.
X0 is the horizontal distance traveled while the cyclist and bicycle are in the air.
m0 is the combined mass of the stunt cyclist and bicycle.
(a)Derive an expression for the distance X0 in terms of H0 , m0, and physical constants, as appropriate.
(b)If the vertical distance between the top of the first ramp and the launch point were 2H0 instead of H0, with no other changes to the first ramp, what is the maximum number of cars that the stunt cyclist could jump over?Justify your answer, using the expression you derived in part (a).
(c)On the axes below, sketch a graph of the vertical component of the stunt cyclist’s velocity as a function oftime from immediately after the cyclist leaves the ramp to immediately before the cyclist lands on the secondramp. On the vertical axis, clearly indicate the initial and final vertical velocity components in terms of H0, θ0, m0 , and physical constants, as appropriate. Take the positive direction to be upward.
2.(12 points, suggested time 25 minutes)
A group of students is investigating how the thickness of a plastic rod affects the maximum force Fmax with which the rod can be pulled without breaking. Two students are discussing models to represent how Fmax depends on rodthickness.
Student A claims that Fmax is directly proportional to the radius of the rod.
Student B claims that Fmax is directly proportional to the cross-sectional area of the rod—the area of the base of thecylinder, shaded gray in the figure above.
(a)The students have a collection of many rods of the same material. The rods are all the same length but comein a range of six different thicknesses. Design an experimental procedure to determine which student’s model,if either, correctly represents how Fmax depends on rod thickness.
In the table below, list the quantities that would be measured in your experiment. Define a symbol to represent each quantity, and also list the equipment that would be used to measure each quantity. You do not need to fill in every row. If you need additional rows, you may add them to the space just below the table.
Describe the overall procedure to be used, referring to the table. Provide enough detail so that another student could replicate the experiment, including any steps necessary to reduce experimental uncertainty. As needed, use the symbols defined in the table and/or include a simple diagram of the setup.
(b)For a rod of radius r0, it is determined that Fmax is F0, as indicated by the dot on the grid below. On thegrid, draw and label graphs corresponding to the two students’ models of the dependence of Fmax on rodradius. Clearly label each graph “A”or“B,” corresponding to the appropriate model.
The table below shows results of measurements taken by another group of students for rods of different thicknesses.
(c)On the grid below, plot the data points from the table. Clearly scale and label all axes, including units.Draw either a straight line or a curve that best represents the data.
(d)Which student’s model is more closely represented by the evidence shown in the graph you drew inpart (c) ?
____Student A’s model: Fmax is directly proportional to the radius of the rod.
____Student B’s model: Fmax is directly proportional to the cross-sectional area of the rod.
Explain your reasoning.