50. A spring in a toy car is compressed a distance, x. When released, the spring returns to its original length, transferring its energy to the car. Consequently, the car having mass m moves with speed v.
Assuming an ideal mechanical system with no loss of energy, derive the spring constant, k, of the car’s spring in terms of m, x, and v. Show all work, including the equations used to derive the spring constant.

Base your answers to questions 51 and 52 on the information below.
A 75kilogram athlete jogs 1.8 kilometers along a straight road in 1.2 × 10^{3} seconds.
51. Determine the average speed of the athlete in meters per second. 
Base your answers to questions 53 and 54 on the information below.
A copper wire at 20°C has a length of 10.0 meters and a crosssectional area of 1.00 × 10^{–3} meter^{2}. The wire is stretched, becomes longer and thinner, and returns to 20°C.
53. What effect does this stretching have on the wire’s resistance? 
Base your answers to questions 56 and 57 on the information and diagram below. Two plane mirrors are positioned perpendicular to each other as shown. A ray of monochromatic red light is incident on mirror 1 at an angle of 55°. This ray is reflected from mirror 1 and then strikes mirror 2.
56. Determine the angle at which the ray is incident on mirror 2. 
Base your answers to questions 58 and 59 on the information and diagram below.
A soccer ball is kicked from point P_{i} at an angle above a horizontal field. The ball follows an ideal path before landing on the field at point P_{f} .
58. On the diagram in your answer booklet, draw an arrow to represent the direction of the net force on the ball when it is at position X. Label the arrow
F_{net}. [Neglect friction.]

62. A tau lepton decays into an electron, an electron antineutrino, and a tau neutrino, as represented in the reaction below.
On the equation in your answer booklet, show how this reaction obeys the Law of Conservation of Charge by indicating the amount of charge on each particle. 
