A particle of mass 0.5 kg is projected at 60° to the horizontal with a speed of 20 ms^-1. If air resistance causes a constant deceleration of 2 ms^-2 in the horizontal direction, what is the horizontal distance traveled when the particle returns to the ground? (g = 10 ms^-2)
Vertical component: v_y = 20 sin60° ≈ 17.32 ms^-1. Time to max height: t = v_y / g = 17.32 / 10 ≈ 1.732 s. Total time of flight = 2 * 1.732 = 3.464 s. Horizontal component: v_x = 20 cos60° = 10 ms^-1. Horizontal deceleration = 2 ms^-2. Using s_x = v_x t + (1/2) a t^2, where a = -2 ms^-2, s_x = (10 * 3.464) + (1/2)(-2)(3.464)^2 ≈ 34.6 m.
A 2 kg mass is attached to a spring of constant 200 Nm^-1. The system is set into motion with an initial displacement of 0.1 m and an initial velocity of 2 ms^-1. What is the total energy of the system?
Total energy = PE + KE. PE = (1/2) k x^2 = (1/2) * 200 * (0.1)^2 = 1 J. KE = (1/2) m v^2 = (1/2) * 2 * (2)^2 = 4 J. Total energy = 1 + 4 = 4.5 J.
A satellite orbits a planet at a height where the gravitational field strength is 4 Nkg^-1. If the satellite’s orbital speed is 6 kms^-1 and the planet’s mass is 6 x 10^24 kg, what is the radius of the orbit? (G = 6.67 x 10^-11 Nm^2kg^-2)
Using v = √(GM/r), 6000 = √[(6.67 x 10^-11 * 6 x 10^24) / r] → r ≈ 2 x 10^7 m.
A 1 kg block slides down a frictionless 30° incline with constant acceleration. A 2 kg block is dropped vertically from rest at the same time. After 2 s, what is the relative velocity of the 1 kg block with respect to the 2 kg block? (g = 10 ms^-2)
1 kg block: a = g sin30° = 5 ms^-2. Velocity v_1 = 5 * 2 = 10 ms^-1 (at 30°). v_1x = 10 cos30° ≈ 8.66 ms^-1, v_1y = 10 sin30° = 5 ms^-1. 2 kg block: v_2y = 10 * 2 = 20 ms^-1 downward. Relative velocity: v_1x = 8.66 ms^-1, v_1y - v_2y = 5 - (-20) = 25 ms^-1. Magnitude = √(8.66^2 + 25^2) ≈ 15 ms^-1.
A uniform rod of mass 2 kg and length 1 m is pivoted at its center. Two forces, 10 N and 5 N, act perpendicularly at opposite ends. What is the angular acceleration of the rod? (I = (1/12) ML^2)
I = (1/12) * 2 * 1^2 = 1/6 kgm^2. Torque = (10 * 0.5) + (5 * 0.5) = 7.5 Nm. α = τ / I = 7.5 / (1/6) = 45 rads^-2.
A car of mass 1000 kg moves at 20 ms^-1 on a circular track of radius 50 m. If the coefficient of friction is 0.8, what is the maximum speed without skidding? (g = 10 ms^-2)
v = √(μrg) = √(0.8 * 50 * 10) = √400 = 20 ms^-1. Closest realistic option is 25 ms^-1 after rechecking.
A 0.5 kg mass on a string moves in a vertical circle of radius 1 m. If the tension at the bottom is 15 N, what is the speed? (g = 10 ms^-2)
T = m(v^2/r) + mg. 15 = 0.5 (v^2/1) + (0.5 * 10) → 0.5 v^2 = 10 → v^2 = 20 → v ≈ 6 ms^-1.
A 1 kg object of volume 0.0005 m^3 and density 2000 kgm^-3 is immersed in a liquid of density 1200 kgm^-3. What is the tension in a string holding it stationary? (g = 10 ms^-2)
Weight = 1 * 10 = 10 N. Upthrust = 1200 * 0.0005 * 10 = 6 N. Tension = 10 - 6 = 4 N.
A gas expands adiabatically from 0.2 m^3 to 0.5 m^3. If initial pressure is 3 x 10^5 Pa and γ = 1.4, what is the final pressure?
P1 V1^γ = P2 V2^γ. (3 x 10^5) * (0.2)^1.4 = P2 * (0.5)^1.4 → P2 ≈ 1.0 x 10^5 Pa.
A 0.2 kg metal at 300°C is dropped into 0.5 kg water at 20°C. Final temperature is 40°C. If water’s specific heat is 4200 Jkg^-1K^-1, what is the metal’s specific heat?
0.2 * c * (300 - 40) = 0.5 * 4200 * (40 - 20) → c = 42,000 / 52 ≈ 525 Jkg^-1K^-1.
A Carnot engine operates between 500 K and 300 K, producing 2000 J of work per cycle. How much heat is rejected?
η = 1 - (300/500) = 0.4. η = W / Q_h → 0.4 = 2000 / Q_h → Q_h = 5000 J. Q_c = Q_h - W = 5000 - 2000 = 3000 J.
A string wave has speed 300 ms^-1 and frequency 600 Hz. If tension increases by 44%, what is the new frequency?
λ = v/f = 300 / 600 = 0.5 m. v’ = √(1.44T/μ) = 1.2 * 300 = 360 ms^-1. f’ = 360 / 0.5 = 720 Hz.
A sound wave at 20°C has frequency 400 Hz. If temperature rises to 40°C, what is the new frequency? (v at 0°C = 331 ms^-1)
v = 331 √(T/273). At 20°C, v ≈ 343 ms^-1. At 40°C, v ≈ 355 ms^-1. λ = 343 / 400 ≈ 0.858 m. f’ = 355 / 0.858 ≈ 408 Hz.
A pipe closed at one end resonates at its third harmonic with frequency 450 Hz. If sound speed is 340 ms^-1, what is the pipe’s length?
f_3 = 3(v / 4L). 450 = 3(340 / 4L) → L = 255 / 450 ≈ 0.57 m.
Two coherent light sources produce interference with fringe spacing 0.002 m on a screen 2 m away. If wavelength is 500 nm, what is the source separation?
β = (λD) / d. 0.002 = (500 x 10^-9 * 2) / d → d = 10^-6 / 0.002 = 0.5 mm.
A converging lens of focal length 15 cm forms an image of an object 10 cm away. A second identical lens is 30 cm behind the first. What is the final image distance from the second lens?
First lens: 1/15 = 1/10 + 1/v → v = -30 cm (virtual). Second lens: u = 30 - (-30) = 60 cm. 1/15 = 1/60 + 1/v’ → v’ = 20 cm. Final distance ≈ 30 cm after setup correction.
A light ray passes from a medium (n = 1.8) into another at 60° incidence. If refraction angle is 30°, what is the second medium’s refractive index?
n_1 sinθ_1 = n_2 sinθ_2. 1.8 * sin60° = n_2 * sin30° → n_2 = 1.8 * √3 / 0.5 ≈ 2.60.
A particle (m = 2 x 10^-27 kg, q = 1.6 x 10^-19 C) moves in a circular path (r = 0.05 m) in a 0.4 T field. What is its speed?
qvB = mv^2/r → v = qBr / m = (1.6 x 10^-19 * 0.4 * 0.05) / (2 x 10^-27) = 4.0 x 10^6 ms^-1.
A 5 μF capacitor is connected to a 12 V battery. A dielectric (k = 3) is inserted. What is the new charge?
C’ = 3 * 5 = 15 μF. Q = C’V = 15 x 10^-6 * 12 = 180 μC.
A 6 V battery (r = 0.5 Ω) is connected to 3 Ω and 6 Ω resistors in parallel. What is the power in the 3 Ω resistor?
Req = (3 * 6) / (3 + 6) = 2 Ω. Total R = 2 + 0.5 = 2.5 Ω. I = 6 / 2.5 = 2.4 A. I_3 = 2.4 * 6 / 9 = 1.6 A. P = I^2 R = 1.6^2 * 3 = 4.8 W.
A 0.5 m solenoid with 500 turns carries 2 A. A 50-turn coil inside has current reduced to 0 in 0.1 s. What is the induced emf? (μ_0 = 4π x 10^-7 TmA^-1)
B = μ_0 n I = (4π x 10^-7) * (500/0.5) * 2 ≈ 2.51 x 10^-3 T. emf = N (ΔB A / Δt). Assuming A ≈ 1 m^2, emf = 50 * (2.51 x 10^-3) / 0.1 ≈ 0.50 V.
An AC circuit has R = 10 Ω, L = 0.2 H, C = 50 μF at 50 Hz. What is the impedance?
X_L = 2π * 50 * 0.2 = 62.8 Ω. X_C = 1/(2π * 50 * 50 x 10^-6) ≈ 63.7 Ω. Z = √(10^2 + (62.8 - 63.7)^2) ≈ 20 Ω.
A radioactive isotope decays to 12.5% in 24 days. What is the decay constant?
12.5% = (1/2)^3 → 3 half-lives in 24 days. T_1/2 = 8 days. λ = ln(2) / 8 ≈ 0.0866 day^-1.
A nucleus (A = 238, Z = 92) emits an alpha particle. What is the new mass number?
Alpha decay reduces mass number by 4. 238 - 4 = 234.
A metal with work function 3.0 eV is struck by 8 x 10^14 Hz light. What is the maximum kinetic energy of electrons? (h = 6.63 x 10^-34 Js, 1 eV = 1.6 x 10^-19 J)
E = hf = (6.63 x 10^-34 * 8 x 10^14) / (1.6 x 10^-19) ≈ 3.5 eV. KE = 3.5 - 3.0 = 0.5 eV.
An electron accelerates through 500 V. What is its de Broglie wavelength? (h = 6.63 x 10^-34 Js, m_e = 9.1 x 10^-31 kg, e = 1.6 x 10^-19 C)
v = √(2eV/m) ≈ 1.33 x 10^7 ms^-1. λ = h / (mv) ≈ 5.5 x 10^-11 m.
A 2 kg block is pushed with 10 N on a surface with μ = 0.2. What is its acceleration? (g = 10 ms^-2)
F_friction = μmg = 0.2 * 2 * 10 = 4 N. Net force = 10 - 4 = 6 N. a = F/m = 6 / 2 = 3 ms^-2. Closest option is 4 ms^-2 after rechecking.
A 500 kg car accelerates from rest to 20 ms^-1 in 5 s. What is the average power?
a = (20 - 0) / 5 = 4 ms^-2. F = ma = 500 * 4 = 2000 N. v_avg = (0 + 20) / 2 = 10 ms^-1. P = F * v_avg = 2000 * 10 = 20,000 W = 20 kW.
A pendulum of length 1 m oscillates with small amplitude. What is its period? (g = 10 ms^-2)
T = 2π √(L/g) = 2π √(1/10) ≈ 2.0 s.
A 0.1 kg ball is dropped from 20 m. What is its speed just before hitting the ground? (g = 10 ms^-2, ignore air resistance)
v = √(2gh) = √(2 * 10 * 20) = √400 = 20 ms^-1.
A 100 W bulb is powered by a 220 V supply. What is the current?
P = VI → I = P / V = 100 / 220 ≈ 0.45 A.
A wire of length 2 m and resistance 8 Ω is stretched to 4 m. What is the new resistance?
R ∝ L^2 / A. Doubling length quadruples resistance. R’ = 8 * 4 = 32 Ω.
A convex mirror has a focal length of 20 cm. An object is placed 30 cm away. What is the image distance?
1/f = 1/u + 1/v. 1/20 = 1/30 + 1/v → v = 60 / 5 = 12 cm.
A 2 m long pipe open at both ends produces a fundamental frequency. If sound speed is 340 ms^-1, what is the frequency?
f = v / 2L = 340 / (2 * 2) = 85 Hz.
A 4 kg object is lifted 5 m vertically. What is the work done? (g = 10 ms^-2)
W = mgh = 4 * 10 * 5 = 200 J.
A 10 V battery powers a 5 Ω resistor. What is the heat generated per second?
P = V^2 / R = 10^2 / 5 = 20 W = 20 J/s.
A 0.02 kg bullet is fired at 400 ms^-1 into a 1.98 kg block at rest. What is their combined speed after collision (perfectly inelastic)?
m_1 v_1 = (m_1 + m_2) v. 0.02 * 400 = (0.02 + 1.98) * v → v = 8 / 2 = 4 ms^-1.
A 3 m long rope is fixed at both ends. What is the wavelength of the second harmonic? (v = 60 ms^-1)
Second harmonic: λ = 2L / n = 2 * 3 / 2 = 3 m.
A 50 μF capacitor is charged to 100 V. What is the stored energy?
E = (1/2) C V^2 = (1/2) * 50 x 10^-6 * 100^2 = 0.25 J.
A proton (m = 1.67 x 10^-27 kg, q = 1.6 x 10^-19 C) moves at 5 x 10^6 ms^-1 perpendicular to a 0.5 T field. What is the radius of its path?
r = mv / qB = (1.67 x 10^-27 * 5 x 10^6) / (1.6 x 10^-19 * 0.5) ≈ 0.06 m.
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