Questions & explanations
1. Rain is falling vertically with a speed of 30 m/s. A man runs horizontally with a speed of 10 m/s. What is the magnitude of the relative velocity of rain with respect to the man?
- 20√2 m/s
- 10√10 m/s
- 20 m/s
- 40 m/s
Answer: 10√10 m/s
Take the downward direction as positive vertical (j) and the man's running direction as positive horizontal (i). Velocity of rain v_r = 30 j m/s, velocity of man v_m = 10 i m/s. The relative velocity of rain w.r.t. man is v_rm = v_r - v_m = -10 i + 30 j. Its magnitude = √(10² + 30²) = √(100+900) = √1000 = 10√10 m/s. Option A (20√2) arises from incorrectly adding 30 and 10 then taking square root; Option C (20) subtracts the speeds; Option D (40) simply adds the speeds, which ignores the vector nature.
2. A projectile is launched with a speed of 10√2 m/s at an angle of 45° to the horizontal. What is the magnitude of its velocity after 1 second? (Take g = 10 m/s²)
- 10 m/s
- 10√2 m/s
- 20 m/s
- 0 m/s
Answer: 10 m/s
The horizontal component of velocity remains constant at u cosθ = 10√2 × cos45° = 10√2 × (1/√2) = 10 m/s. The vertical component initially is u sinθ = 10√2 × sin45° = 10 m/s. After 1 second, vy = u sinθ - g t = 10 - 10×1 = 0 m/s. Hence the velocity at t=1 s is purely horizontal with magnitude 10 m/s. Option B would be the initial speed, Option C is twice the horizontal component and Option D would require zero vertical and horizontal velocity, which does not occur.
3. Which of the following correctly distinguishes non‑uniform circular motion from uniform circular motion?
- In non‑uniform circular motion, the speed changes with time; in uniform circular motion, it remains constant.
- In uniform circular motion, there is a tangential acceleration; in non‑uniform circular motion, there is none.
- Both have zero centripetal acceleration.
- In non‑uniform circular motion, the centripetal acceleration is constant in magnitude; in uniform circular motion, it varies.
Answer: In non‑uniform circular motion, the speed changes with time; in uniform circular motion, it remains constant.
Uniform circular motion has constant speed, so only centripetal acceleration exists. Non‑uniform circular motion has changing speed, so tangential acceleration appears in addition to centripetal acceleration. Option B is reversed. Option C is false because both have centripetal acceleration. Option D is false because in uniform circular motion centripetal acceleration magnitude is constant (v²/r), while in non‑uniform motion it may vary.
4. A stone is thrown horizontally with a speed of 10 m/s from the top of a tower of height 5 m. What is the magnitude of its velocity just before hitting the ground? (Take g = 10 m/s²)
- 10√2 m/s
- 20 m/s
- 10 m/s
- 5√2 m/s
Answer: 10√2 m/s
Time of flight t = √(2h/g) = √(2×5/10) = √1 = 1 s. Vertical velocity gained vy = g t = 10×1 = 10 m/s. Horizontal velocity remains constant at 10 m/s. Resultant speed = √(10² + 10²) = 10√2 m/s. Option B (20 m/s) is incorrect; it would be the vertical speed if the time were 2 s. Option C (10 m/s) ignores the vertical component. Option D (5√2 m/s) uses the height incorrectly.
5. For one-dimensional motion along a straight line, which of the following statements is correct?
- Distance is always equal to displacement.
- Speed is a vector quantity, while velocity is a scalar.
- Displacement can be zero even if the distance travelled is non-zero.
- Velocity is always positive in one-dimensional motion.
Answer: Displacement can be zero even if the distance travelled is non-zero.
Displacement is the change in position; if the object returns to its starting point, displacement is zero while the distance (total path length) is non-zero. The other options are false: distance is always greater than or equal to displacement, speed is scalar and velocity is vector, and velocity can be negative depending on direction.
6. A particle moves in a circular path with uniform speed. Which of the following statements about its motion is correct?
- The velocity of the particle is constant.
- The speed of the particle is constant, but the velocity changes continuously.
- The acceleration of the particle is zero.
- Both the speed and the velocity of the particle are constant.
Answer: The speed of the particle is constant, but the velocity changes continuously.
In uniform circular motion, the particle travels at a constant speed (magnitude of velocity). However, the direction of velocity changes at every point along the curved path, so velocity (a vector) is not constant. The change in velocity gives rise to a centripetal acceleration, which is non-zero and directed toward the center.
7. Angular velocity of a particle in circular motion is defined as the:
- rate of change of angular displacement
- rate of change of linear displacement
- rate of change of angular acceleration
- product of radius and linear velocity
Answer: rate of change of angular displacement
Angular velocity is the time rate of change of angular displacement, measured in rad/s. The other options are incorrect: rate of change of linear displacement is linear velocity, rate of change of angular acceleration is angular jerk, and product of radius and linear velocity does not define angular velocity.
8. A particle moves on a circular path with non‑uniform speed. At a given instant, its centripetal acceleration is 4 m/s² and its tangential acceleration is 3 m/s². The magnitude of the net acceleration is:
- 5 m/s²
- 7 m/s²
- 1 m/s²
- 12 m/s²
Answer: 5 m/s²
In non‑uniform circular motion, the net acceleration is the vector sum of centripetal and tangential components, which are perpendicular. So magnitude = √(a_c² + a_t²) = √(4² + 3²) = 5 m/s². The other options result from incorrect addition (7 from simple sum, 1 from difference, 12 from product).
9. In uniform circular motion, the centripetal acceleration acts:
- along the tangent to the path
- radially outward
- radially inward
- in the direction of velocity
Answer: radially inward
Centripetal acceleration always points towards the center of the circular path, i.e., radially inward. It is perpendicular to the velocity. Options A and D describe the direction of velocity, and option B is the direction of the fictitious centrifugal force, not the actual acceleration.
10. The velocity-time graph of a body is a straight line with a positive slope passing through the origin. This implies that the body has:
- constant velocity
- constant acceleration
- decreasing acceleration
- zero displacement at any time
Answer: constant acceleration
The slope of a velocity-time graph gives acceleration. A straight line with constant positive slope means acceleration is constant and positive. Constant velocity would be a horizontal line. Acceleration is not decreasing. Displacement (area under curve) is non-zero for t > 0.
11. A person walks 3 km east in 1 hour, then 4 km west in 2 hours. What are the average speed and average velocity for the entire trip?
- Average speed = 7/3 km/h, average velocity = -1/3 km/h
- Average speed = 7/3 km/h, average velocity = 1/3 km/h
- Average speed = 7/3 km/h, average velocity = -1 km/h
- Average speed = 3 km/h, average velocity = 0 km/h
Answer: Average speed = 7/3 km/h, average velocity = -1/3 km/h
Total distance = 3 km + 4 km = 7 km, total time = 3 h, so average speed = 7/3 km/h. Displacement = 3 km (east) - 4 km (west) = -1 km (west), so average velocity = -1/3 km/h. The negative sign indicates direction (west). Other options have incorrect magnitudes or sign.
12. The position-time graph of a particle moving along the x-axis is a straight line with a positive slope. Which of the following is correct about the motion?
- The particle is at rest.
- The particle has a constant negative velocity.
- The particle has a constant positive velocity.
- The particle is accelerating.
Answer: The particle has a constant positive velocity.
A straight line on a position-time graph indicates constant velocity. A positive slope means velocity is positive (increasing position with time). Rest would be a horizontal line, negative velocity would be a negative slope, and acceleration would be a curved graph.