31. For an ideal gas in thermal equilibrium, which of the following statement is true?

(We define v_av, v_mp, and v_rms to be the average, the most probable, and the root-mean-square speeds of the gas respectively.)
(A) v_av ＜ v_mp ＜ v_rms
(B) v_mp ＜ v_av ＜ v_rms
(C) v_rms ＜ v_av ＜ v_mp
(D) v_rms ＜ v_mp ＜ v_av
(E) None of the above. 

1. Given the velocity of a car in two dimensions: = (t²-1, 3). The car starts out at position (x, y) = (2, 4) at t = 0. What is the car at t = 3? (A) (6,9) (B) (8,3) (C) (10,7) (D) (8,13) (E) none of the above.

2. An airplane flies at an altitude of 0.5 km and a speed of 360 km/h. At what horizontal distance x should it release a bomb to hit a target that is moving with constant velocity 90 km/h? (Approximate the gravitational acceleration g by 10 m/s² throughout this test) (A) 500 (B) 750 (C) 250 (D) 1000 (E) none of the above.

3. A boy wishes to row across a river in the shortest possible time. He can row at 2 m/s in still water and the river is flowing at 1 m/s. At what angle with respect to the water velocity should he point the bow (the front of the boat)? (A) 30° (B) 45° (C) 60° (D) 90° (E) none of the above.

4. Two blocks of mass 25 kg and 75 kg are connected by a light rope across a fixed pulley of radius 50 cm and moment of inertia I = 10 kg·m². Assume that the rope never slips. What is the magnitude of the acceleration of the 75 kg block? (A) 0.54 m/s² (B) 0.1 m/s² (C) 0.049 m/s² (D) 0.8 m/s² (E) none of the above.

5. A rope is used to pull a 10 kg box with constant velocity across the floor with coefficient of kinetic friction μk = 1/√3. At what angle (in degrees) is the force F least? (A) 30 (B) 45 (C) 60 (D) 0 (E) none of the above.

6. My weight is mg and the maximum speed my jump is v₀. If I jump off a cart of mass M that can move without friction, what is my maximum horizontal displacement via adjusting the angle of my jump? Note that v₀ is now measured relative to the car. (A) (B) (C) (D) (E) none of the above.

7. An object of mass m that is initially at rest slides down from the top of a smooth sphere (of radius R). If the size of the object is negligible, at what angle θ will it detach from the sphere? (A) cosθ = 1/2 (B) sinθ = 1/2 (C) sinθ = 2/3 (D) cosθ = 2/3 (E) none of the above.

8. A uniform soda can of base area A = 12 cm², mass m = 10 g and height h = 20 cm is filled with soda of density ρ = 1 g/cm³. Use x to denote the height of the remaining soda; find the x that makes the center of mass of the whole system the lowest. It occurs when the soda is (A) half-filled (B) less than half, but more than a quarter filled (C) less than a quarter filled (D) completely empty (E) none of the above.

9. A 10 kg monkey climbs up a light rope that runs over a frictionless tree limb and back down to a 20 kg package on the ground. What is the least acceleration the monkey must have if it is to lift the package off the ground? (A) g/4 (B) g/2 (C) g (D) 2g (E) none of the above.

10. A block of mass m = 10 kg slides from vertical height h = 1 m down a slope of angle θ = π/3 and kinetic friction coefficient μk = 0.5. At the foot of the slope lies a spring of constant K = 200 nt/m. What is the maximum compression of the spring? (A) 1 (B) 1.5 (C) 2 (D) 2.5 (E) none of the above.

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