8. Suppose the velocity of a falling sphere of mass m due to the gravity and the air resistance is given by v(t) = v₁(1-e⁻ᵞᵗ) with the condition that the air resistance vanishes when v = 0. The formula of the air drag force is then given by
(A) -2mγv
(B) -mγv
(C) -mγv²
(D) -mγv²/2.

4. A billiard ball hits another ball at rest of identical mass on a surface, making the latter to leave at an angle of 37° relative to the incident direction of the first ball. Suppose the incident speed is u. What is the speed of the first ball after collision?(A) 0.36u(B) 0.6u(C) 0.64u(D) 0.8u.

5. A bomb explodes and breaks into two pieces of mass ratio 2:1. Neglect the mass of the explosive chemicals. What is the ratio of the kinetic energy obtained by the two pieces?(A) 1:2(B) 2:1(C) 4:1(D) 1:3.

6. Consider a solar system with a sun of mass two times of our sun. A planet orbiting this sun has a period of one earth year. What is the distance from the sun to the planet? (Hint: au is the average distance from the earth to our sun.)(A) 2⁻¹/² au(B) 2⁻¹/³ au(C) 2¹/² au(D) 2¹/³ au.

7. A sphere of mass m and another sphere of mass 2m are connected by a string of length R. Suppose they are now rotating about their center of mass at an angular velocity ω. Suddenly the string breaks. What is the velocity of the massive one (2m) after the string breaks?(A) Rω(B) Rω/2(C) Rω/3(D) 2Rω/3.

9. Which of the following statements is not compatible with other 3 statements? (A) It is impossible for a heat engine that operates in a cycle to convert its heat input completely into work. (B) It is impossible for a cyclical device to transfer heat continuously from a cold body to a hot one without the input of work or other effect on the environment. (C) The change in the internal energy of the system comes from the heat input minus the work done on its surroundings. (D) The entropy change of an isolated system always stays constant or increases.

10. The rate of heat flow through a slab is R. If the slab thickness is doubled, its cross-sectional area is halved, and the temperature difference across it is doubled, then the rate of heat flow becomes (A) 2R (B) R/2 (C) 8R (D) R/8.

11. A molar monatomic ideal gas does adiabatic free expansion from an initial state of pressure P and volume V/2 to volume 2V, and then raise the pressure to 2P while keeping constant volume. Find the change of entropy: (A)Rln2(B)Rln2(C)Rln 2 (D) Rln 2.

12. The temperature of low-pressure hydrogen is reduced from 127°C to 47°C. The rms speed of its molecules decreases by approximately (A) 5% (B) 10% (C) 15 % (D) 20%.

13. Which of the following statements about the Carnot engine is not correct? (A) The efficiency of a Carnot engine operating between temperatures T₁ and T2 (T₁ ＞ T2) is 1 – T2/T1. (B) The efficiency of any reversible engine is identical to that of a Carnot engine for given two heat reservoirs. (C) A Carnot engine has the best efficiency for given two heat reservoirs. (D) The entropy of a Carnot engine during a cycle remains constant.

14. Consider a partial circuit shown in the figure. Suppose every resistor is of identical resistance R. Find the effective resistance between A and B: (A) R (B) R/2 (C) 2R/3 (D) R/6.