31. A particle moves along the x-axis with an acceleration given by a(t) = -kv², where v is the instantaneous velocity and k is a positive constant. If the particle has an initial velocity v₀ at x = 0, what is the velocity as a function of position x?
(A) v(x) = v₀ - kx
(B) v(x) = v₀e⁻ᵏˣ
(C) v(x) =
(D) v(x) = v₀/(1 + kx)
(E) v(x) = v₀ - kx²

32. A uniform circular disk of mass M and radius r is connected by a light cord to a fixed pivot on a smooth horizontal table. The center of the disk moves at a constant distance R from the pivot and undergoes uniform circular motion with tangential speed v. The magnitude of the disk's angular momentum about the pivot is (A) MRv (B) (C) (D) (E) M(R² + r²/2)(v/R)

33. A 20 g bullet moving vertically upward at 1000 m/s strikes and passes through the center of mass of a 4.0 kg block initially at rest. The bullet emerges from the block moving directly upward at 400 m/s. Neglecting air resistance, to what maximum height does the block then rise above its initial position? (Gravitational acceleration g = 10 m/s²) (A) 0.3 cm (B) 2.1 cm (C) 15 cm (D) 30 cm (E) 45 cm

34. A water storage tank with cross-sectional area A₁ is filled to a height of h. The space above the water contains air at pressure 3p₀. The water flows out the bottom of the tank through a short pipe with cross-sectional area A₂. The area A₁ is much larger than A₂. The pressure at the exit pipe is p₀. The water with density ρ is considered an ideal fluid. Find the flow speed in the pipe. (Gravitational acceleration g) (A) (B)  (C) (D) (E)

35. In hydrotherapy training at KMU, a therapist measures a patient's body density using underwater weighing. A patient has a mass of 75.0 kg in air. When fully submerged in water, the apparent mass is 3.0 kg. Assume water density ρ = 1.0 g/cm³ and neglect lung air volume changes. What is the patient's body density? (A) 0.960 kg/m³ (B) 1.042 kg/m³ (C) 960 kg/m³ (D) 1042 kg/m³ (E) 1120 kg/m³

36. A hospital keeps oxygen cylinders in an emergency room. An oxygen cylinder contains gas at a gauge pressure of 220 kPa at a temperature of 27 °C. During transportation in summer, the cylinder temperature rises to 47 °C. What is the new gauge pressure inside the cylinder? Assume the cylinder volume is constant, oxygen behaves as an ideal gas, and atmospheric pressure is 100 kPa. (A) 234 kPa (B) 241 kPa (C) 331 kPa (D) 334 kPa (E) 341 kPa

37. During a liver ultrasound scan, the transducer operates at frequency f = 5.0 MHz. The sound speed in liver tissue is v = 1540 m/s. The emitted ultrasound pulse contains 3 cycles per pulse (i.e., the spatial pulse length (SPL) corresponds to 3 wavelengths). Assume the minimum separable distance along the beam direction (axial resolution) is Δzmin = SPL/2. What is the minimum tumor thickness (along the beam direction) that can be resolved as a distinct structure? (A) 0.15 mm (B) 0.23 mm (C) 0.46 mm (D) 0.92 mm (E) 1.54 mm

38. The deformation and the restoring force of an elastic string obey Hooke's law. When the string is stretched to twice its original length, the speed of transverse waves on the string is v. What is the wave speed on the string when it is stretched to four times its original length? Assume the mass of the string remains constant and is uniformly distributed. (A) v (B) (C) (D) 2v (E)

39. A supersonic jet fighter flies horizontally at a constant altitude of 3400 m with a speed of Mach 1.25. Assuming the speed of sound is 340 m/s, how much time will elapse after the jet passes directly over an observer before the shock wave (sonic boom) reaches them? (A) 4.00 s (B) 6.00 s (C) 7.50 s (D) 8.00 s (E) 10.0 s

40. An ideal monatomic gas expands from volume V₀ to 2V₀ along a straight line path in the P-V (pressure-volume) diagram from (P₀, V₀) to (P₀/2, 2V₀). What is the heat Q absorbed by the gas in this process? (A) 0.75P₀V₀ (B) P₀V₀ (C) 1.25P₀V₀ (D) 1.5P₀V₀ (E) 2P₀V₀

41. A cylindrical rod of length L and radius R is insulated on its surface. Its thermal conductivity k varies with temperature as k(T) = αT. If the two ends of the rod are maintained at T₁ and T₂, the heat current dQ/dt is proportional to ______. (A) ln(T₁/T₂) (B) (C) T₁ - T₂ (D) T₁² - T₂² (E) T₁³ - T₂³

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