Question

A toy car is pressed against a horizontal spring, compressing the spring from its relaxed length. The car is then released, and the spring launches the car horizontally along a track that ends in a ramp, as shown. The car has enough speed to leave the ramp, and the car reaches a maximum vertical height above the floor. Assume there is negligible friction between the car and the track and air resistance is negligible.

• θ (theta) is the angle of the ramp from the horizontal.

• h is the maximum height reached by the car.

• k is the spring constant of the spring.

• x is the distance the spring is compressed.

• M is the mass of the car

A second trial is conducted using a spring with a spring constant that has a greater value than that of the original spring. The new spring is compressed the same distance x, and the same car is used.

(a) Indicate how using the new spring would affect the car’s maximum height, and explain why without mathematically deriving a relation for h.

Two students are discussing the motion of the car when it is at its maximum height and make the following statements:

Student 1: At the car’s maximum height, its speed is zero, because the kinetic energy is a minimum at the top of the trajectory.

Student 2: Only the vertical component of the velocity is zero at the maximum height, because the vertical component of the acceleration is zero at the top of the trajectory.

(b) Student 1 Analysis:

1. Which underlined phrase or phrases in student 1’s statement are correct, if either? If neither phrase is correct, write “none.” Justify your answer.

2. Which underlined phrase or phrases in student 1’s statement are incorrect, if either? If neither phrase is incorrect, write “none.” Justify your answer

(c). Student 2 Analysis:

1. Which underlined phrase or phrases in student 2’s statement are correct, if either? If neither phrase is correct, write “none.” Justify your answer.

2. Which underlined phrase or phrases in student 2’s statement are incorrect, if either? If neither phrase is incorrect, write “none.” Justify your answer

4. (d)  Consider the system consisting of the car and the spring, but not Earth. How does the energy of the system when the spring is fully compressed compare to the energy of the system at the moment immediately before the car hits the ground? Justify your answer.

5. (e)  If the friction between the track and the car were not negligible, state one way in which the setup could be changed such that the maximum height of the car was the same as in the original setup. Justify your answer.