Motion — Set 6

Correct Answer: B. a = v^2/r

• **a = v²/r** = Centripetal acceleration in uniform circular motion equals the square of the speed divided by the radius. • **a = v²/r = ω²r** — directed toward the center; larger speed or smaller radius gives greater centripetal acceleration. • 💡 Wrong-option analysis: a = r/v²: inverts both numerator and denominator — dimensionally incorrect (units would be s²/m not m/s²); a = v/r²: wrong formula — gives wrong units; a = r²/v: dimensionally inconsistent.

Correct Answer: C. v = rω

• **v = rω** = Linear speed v equals the radius r multiplied by the angular speed ω. • **v = rω** — derived from arc length s = rθ; differentiating: v = ds/dt = r × dθ/dt = rω. • 💡 Wrong-option analysis: v = ω/r: inverts r — gives units of (rad/s)/m = 1/(m·s), not m/s; v = r/ω: inverts ω — gives units of m/(rad/s) = m·s, not m/s; v = r²ω: extra factor of r — gives units of m²/s, not m/s.

Correct Answer: C. Radian

• **Radian** = The SI unit of angular displacement is the radian (rad), defined as arc length divided by radius. • **1 rad = arc length/radius** — one complete revolution = 2π rad ≈ 6.28 rad; radian is a dimensionless SI derived unit. • 💡 Wrong-option analysis: Revolution: a practical unit for full cycles but is not the SI unit of angle; Meter: meter is the SI unit of linear distance, not angle; Degree: widely used but not the SI unit of angular measurement.

Correct Answer: B. f = 1/T

• **f = 1/T** = Frequency is the number of complete cycles per second; if one cycle takes T seconds, then f = 1/T Hz. • **f = 1/T** — e.g. if T = 0.25 s then f = 4 Hz; SI unit of frequency is hertz (Hz = s⁻¹). • 💡 Wrong-option analysis: f = 2T: dimensionally wrong — 2T has units of seconds, not Hz; f = T²: dimensionally wrong — T² has units of s², not s⁻¹; f = T: implies f in Hz equals T in s, which is dimensionally incorrect.

Correct Answer: B. (2u sinθ)/g

• **(2u sinθ)/g** = Time of flight is determined by vertical motion; the projectile returns to the same level when T = 2u sinθ/g. • **T = 2u sinθ/g** — at 45°, T = 2u×(√2/2)/g = u√2/g; the vertical component u sinθ sets the flight duration. • 💡 Wrong-option analysis: (u cosθ)/g: mixes the horizontal component with g — has units of s but is not the time of flight formula; (u² sin2θ)/g: this is the range formula, not time of flight; (2u cosθ)/g: uses horizontal velocity component instead of vertical — incorrect for time of flight.

Correct Answer: B. (u^2 sin2θ)/g

• **(u² sin2θ)/g** = Range = horizontal velocity × time of flight = u cosθ × 2u sinθ/g = u² sin2θ/g. • **R = u² sin2θ/g** — maximum range at θ = 45° where sin90° = 1; complementary angles give equal range. • 💡 Wrong-option analysis: (u cosθ)/g: gives units of (m/s)/(m/s²) = s, i.e. a time not a distance; (2u sinθ)/g: this is the time of flight, not range; (u sinθ)/g: this is half the time of flight, not range.

Correct Answer: D. 45°

• **45°** = Range R = u² sin2θ/g is maximum when sin2θ = 1, i.e. 2θ = 90° → θ = 45°. • **θ = 45° maximises range** — equal horizontal and vertical components yield the longest flight distance for given speed. • 💡 Wrong-option analysis: 60°: sin120° = sin60° ≈ 0.866 < 1, giving less range than 45°; 30°: sin60° ≈ 0.866, same range as 60° but less than 45°; 90°: sin180° = 0, so range is zero — projectile goes straight up.

Correct Answer: D. vA - vB

• **vA − vB** = The velocity of A relative to B is found by subtracting B's velocity from A's velocity: v_A/B = vA − vB. • **v_A/B = vA − vB** — this gives A's velocity as seen from a frame moving with B. • 💡 Wrong-option analysis: vB − vA: this is the velocity of B relative to A — the reverse; vA + vB: this gives the sum, useful for closing speed of opposite-direction motion but not relative velocity; vA × vB: the cross product gives a vector perpendicular to the plane — not relative velocity.

Correct Answer: A. (u + v)/2

• **(u + v)/2** = For constant acceleration, velocity changes linearly from u to v, so the average is the arithmetic mean: (u + v)/2. • **v_avg = (u+v)/2** — this equals s/t and is valid only for constant (uniform) acceleration. • 💡 Wrong-option analysis: u + v: this is the sum of initial and final velocities, not their average; u − v: this is the change in velocity (with sign reversed) — a vector difference, not average; uv: this is the product of speeds with units m²/s², not a velocity.

Correct Answer: A. Displacement

• **Displacement** = The area under a velocity–time graph equals ∫v dt = total displacement of the object. • **area = ∫v dt = displacement** — positive area = positive displacement; area below time axis = negative displacement. • 💡 Wrong-option analysis: Speed: speed is a point value on the y-axis of the graph, not the area under it; Force: force needs mass × acceleration — not readable as area under a v–t graph; Acceleration: acceleration is the slope of the v–t graph, not the area.