IITJEE-MAINS Physics Previous Year Question Quiz Practice Set 18

Q1. The position vector of an ant (S in metres) moving in the Y-Z plane is given by S(t) = 2 t² ĵ + 5 k̂ (where t is in seconds). What are the magnitude and direction of the instantaneous velocity of the ant at t = 1 s?

Q2. A gas balloon is rising vertically upwards with a constant velocity of 10 m s⁻¹. When the balloon reaches a height of 75 m above the ground, a stone is gently dropped from it while the balloon continues moving upwards at the same constant velocity of 10 m s⁻¹. Taking g = 10 m s⁻², what is the height of the balloon above the ground at the instant when the stone hits the ground?

Q3. A streetcar moves rectilinearly from rest at Station A to come to rest at the next Station B with an instantaneous acceleration varying with distance x from Station A according to the law a(x) = α − β x, where α and β are positive constants. What are the distance between the two stations and the maximum velocity attained by the streetcar?

Q4. The position of a body moving along a straight line is given by x(t) = (−3 t³ + 18 t² + 16 t) m, where t is in seconds. What is the instantaneous velocity of the body at the instant when its instantaneous acceleration becomes zero?

Q5. A solid cylinder rolls off the back of a large truck travelling horizontally at 10 m s⁻¹ to the right. At the instant it leaves the truck, the cylinder is moving horizontally at 8 m s⁻¹ to the left relative to an observer in the truck. If the cylinder lands on the roadway 1.25 m below its starting level (take g = 10 m s⁻²), how far behind the rear of the truck does it land?

Q6. A particle starts from rest at x = 0 and moves with a constant acceleration a along the positive x-axis until it reaches an instantaneous velocity v₀ at position x = L. What is the ratio of its Time-Average Velocity (⟨v⟩_t = ∫v dt / ∫dt) to its Space-Average Velocity (⟨v⟩_x = ∫v dx / ∫dx) over this interval?

Q7. A swimmer can swim with speed u with respect to water. He crosses a river of width d in the shortest possible time (by swimming perpendicular to the banks). The speed of river water is zero at both banks and increases linearly to a maximum value v₀ at the mid-stream of the river, then decreases linearly back to zero at the opposite bank. What is the total downstream drift of the swimmer?

Q8. If the x and y coordinates of a particle moving in a vertical plane as a function of time t (in seconds) are given as x(t) = 24 t and y(t) = 43.6 t − 4.9 t² (in metres), then what is the angle made by the instantaneous velocity vector of the particle with the horizontal at t = 2 s?

Q9. A car is standing 200 m behind a bus, which is also at rest on a straight road. Both start moving forward at the exact same instant (t = 0), with the bus accelerating at a_B = 2 m s⁻² and the car accelerating at a_C = 4 m s⁻². After how much time will the car catch up with the bus, and what is the speed of the car relative to the bus at that instant?

Q10. A particle initially at rest starts moving from reference point x = 0 along the positive x-axis with an instantaneous velocity v that varies with position x as v(x) = 4 √x m s⁻¹. What are the instantaneous acceleration of the particle and its average velocity during the first 3 seconds of motion?

Q11. Two cars A and B are moving in the same direction along a straight highway. Car A (ahead) moves at 100 km h⁻¹ and Car B (behind) moves at 80 km h⁻¹. A person sitting in Car B throws a stone forward with a speed v (relative to Car B) so that it hits Car A with a relative impact speed of 5 m s⁻¹. Neglecting vertical motion and air resistance, what is the value of v in km h⁻¹?

Q12. A particle moves along a circular path of radius R with constant speed v. When the radius vector of the particle sweeps through an angle of θ = 60° (π / 3 rad), what are the magnitude of its displacement (|Δr|) and the magnitude of the change in its velocity vector (|Δv|)?

Q13. A particle moves in the x-y plane with a constant acceleration of a = (1 î − 1 ĵ) m s⁻². If its initial velocity at t = 0 is u = (2 î + 3 ĵ) m s⁻¹, what are the instantaneous speed of the particle at t = 2 s and the magnitude of its displacement during these 2 seconds?

Q14. Two trains of lengths 60 m and 120 m move on parallel tracks with constant speeds of 80 km h⁻¹ and 30 km h⁻¹ respectively. If t₁ is the time taken for them to completely cross each other when moving in the same direction, and t₂ is the time taken when moving in opposite directions, what is the ratio (t₁ / t₂)?

Q15. From a height of 18 m above the ground, a ball is dropped from rest. Neglecting air resistance and taking g = 10 m s⁻², at what height above the ground does the numerical value of its instantaneous velocity become equal to the numerical value of its instantaneous acceleration (in SI units)?