Physics — Work and Energy
Physics — Work and Energy. Practice questions to deepen understanding of physics — work and energy. Online physics practice with full solutions and step-by-step explanations.
Physics work and energy practice — 50 questions: work W = Fd, kinetic and potential energy, conservation of energy, power, efficiency. Formulas and applications.
Part A: Work — questions 1–12.
- Definition of work
Question 1
2.00 pts
⚙️ Work:
What is work in physics?
Explanation:
💡 Detailed explanation:
Work in physics! ⚙️
Work: W = F·d·cos(θ) or in vector notation: W = F⃗·d⃗ (scalar product) 🔍 Components: • F: the force (N) • d: the displacement (m) • θ: the angle between F and d • W: work (J - Joule) 💡 Meaning: Work = energy transfer A force does work → energy is transferred to the system → something changes (velocity, height...) 📊 Properties of work: ✓ Scalar (not a vector!) ✓ Units: Joule (J) = N·m ✓ Can be positive/negative/zero ✓ Depends on the angle θ ⚠️ Important difference from everyday usage: Holding a heavy box without moving = no work in physics! (d = 0) You may sweat, but physically: W = 0 |
Question 2
2.00 pts
⚖️ Units:
What is the unit of work?
Explanation:
💡 Detailed explanation:
Units of work! ⚖️
Joule: 1 J = 1 N·m The work done by a force of 1 Newton over a distance of 1 meter 🔍 Detail: J = N·m J = (kg·m/s²)·m In base units: 1 J = 1 kg·m²/s² 💡 Common conversions: • 1 kJ = 1000 J • 1 MJ = 1,000,000 J • 1 cal ≈ 4.18 J (calorie) • 1 kWh = 3.6 MJ Examples: • Lifting 1 kg by 1 meter ≈ 10 J • Walking 1 km ≈ 240,000 J • 100W bulb for an hour = 360,000 J |
Question 3
2.00 pts
➕➖ Sign of work:
When is work negative?
Explanation:
💡 Detailed explanation:
Sign of work! ➕➖
Depends on cos(θ): W = F·d·cos(θ) 📊 By angle:
💡 Examples: W > 0 (positive): • Pushing a cart in the direction of motion • Engine accelerates a car • Free fall (gravity ↓ + motion ↓) W < 0 (negative): • Friction (always opposes motion) • Brakes (force ← motion →) • Lifting an object (gravity ↓, motion ↑) W = 0 (zero): • Centripetal force (⊥ velocity) • Carrying a box horizontally (gravity ↓, motion →) • Static friction in rolling |
Question 4
2.00 pts
🧮 Exercise:
A force F=20N pushes a box
distance d=5m in the direction of the force
What is the work?
Explanation:
💡 Detailed explanation:
Work calculation! 🧮
| 📐 Solution: Given: F = 20 N d = 5 m θ = 0° (in the direction of the force) Formula: W = F·d·cos(θ) W = 20×5×cos(0°) W = 20×5×1 W = 100 J 💡 Meaning: The force transferred 100 Joules of energy to the box Unit check: [W] = N·m = J ✓ |
Question 5
2.00 pts
🧮 Exercise with angle:
F=50N, d=10m, angle 60°
What is the work?
Explanation:
💡 Detailed explanation:
Work with angle! 🧮
| 📐 Solution: Given: F = 50 N d = 10 m θ = 60° cos(60°) = 0.5 Calculation: W = F·d·cos(θ) W = 50×10×0.5 W = 250 J 💡 Insight: Only half the force is in the direction of motion! F_∥ = F·cos(60°) F_∥ = 50×0.5 = 25N W = 25×10 = 250J ✓ |
Question 6
2.00 pts
❓ Zero work:
When does a force do zero work?
Explanation:
💡 Detailed explanation:
Zero work! ❓
W = 0 when: 1️⃣ θ = 90° cos(90°) = 0 → W = F·d·0 = 0 2️⃣ d = 0 No displacement → W = F·0 = 0 3️⃣ F = 0 No force → W = 0·d = 0 💡 Important examples: Carrying a bag: F↑ (force up) d→ (walking horizontally) θ = 90° → W = 0 No work! (even though it's tiring...) Satellite in orbit: F_c → toward center (radial) v → tangent to the circle F_c ⊥ v always → W = 0 Gravity does no work! (|v| constant) Holding a heavy box: F = mg (force up) d = 0 (not moving) → W = 0 Tiring physiologically but no physical work |
Question 7
2.00 pts
⬇️ Work of gravity:
What is the work done by gravity
when a body m=5kg rises h=10m?
(g=10)
Explanation:
💡 Detailed explanation:
Work of gravity! ⬇️
| 📐 Solution: Force and displacement: • Weight: W = mg = 50N ↓ • Displacement: h = 10m ↑ • Angle: θ = 180° (opposite!) Calculation: W_gravity = F·h·cos(180°) W_gravity = 50×10×(-1) W_gravity = -500 J Negative! 💡 Meaning: Gravity opposes the rise → negative work → gravity removes energy Convenient formula: W_gravity = -mgh (rising) W_gravity = +mgh (falling) or in general: W_gravity = -mg·Δh (Δh positive = rising) ⚠️ Note: When falling: W_gravity > 0 (positive) Gravity helps the motion! |
Question 8
2.00 pts
🔢 Total work:
If several forces act,
what is the total work?
Explanation:
💡 Detailed explanation:
Total work! 🔢
Superposition principle: W_net = W₁ + W₂ + W₃ + ... or: W_net = ΣW_i 💡 Alternative method: First find the net force: F_net = ΣF_i Then: W_net = F_net · d · cos(θ) 📐 Example: A box pushed at F=100N friction f=30N distance d=5m Method 1: sum W_push = 100×5 = 500 J W_friction = -30×5 = -150 J W_net = 500 - 150 = 350 J Method 2: net force F_net = 100 - 30 = 70N W_net = 70×5 = 350 J ✓ Same result! ⚠️ Important: Need to count the signs carefully! Negative work has a negative sign in the algebraic sum. |
Question 9
2.00 pts
⚡ Important theorem:
What is the relation between total work
and the change in kinetic energy?
Explanation:
💡 Detailed explanation:
Work-energy theorem! ⚡
The central theorem: W_net = ΔE_k or in detail: W_net = E_k,final - E_k,initial W_net = ½mv² - ½mv₀² 🔍 Meaning: The total work done on a body equals the change in its kinetic energy In words: • W_net > 0 → speed increases • W_net < 0 → speed decreases • W_net = 0 → speed constant 💡 Example: Car m=1000kg v₀ = 10 m/s v = 20 m/s What is the work done? E_k,0 = ½×1000×10² = 50,000 J E_k = ½×1000×20² = 200,000 J W_net = ΔE_k = 200,000 - 50,000 W_net = 150,000 J = 150 kJ ⭐ Why is this powerful? Don't need to know: • What forces • What trajectory • How much time Only initial and final velocity! |
Question 10
2.00 pts
🧮 Exercise:
Box m=4kg, v₀=0
Force F=20N acts over distance d=10m
What is the final velocity?
Explanation:
💡 Detailed explanation:
Work → velocity! 🧮
| 📐 Full solution: Given: m = 4 kg v₀ = 0 F = 20 N d = 10 m Step 1: Work W = F·d W = 20×10 W = 200 J Step 2: Work-energy theorem W = ΔE_k W = ½mv² - ½mv₀² 200 = ½×4×v² - 0 200 = 2v² v² = 100 v = 10 m/s Verification: E_k,final = ½×4×10² = 200 J ✓ All the work became kinetic energy! 💡 Direct formula: v = √(2W/m) v = √(2×200/4) v = √100 = 10 m/s ✓ |
Question 11
2.00 pts
🧮 Braking:
Vehicle m=1000kg, v₀=20 m/s
brakes to a stop over d=50m
What is the braking force?
Explanation:
💡 Detailed explanation:
Braking force calculation! 🧮
| 📐 Solution: Given: m = 1000 kg v₀ = 20 m/s v = 0 (stop) d = 50 m Step 1: Energy change E_k,0 = ½×1000×20² E_k,0 = 200,000 J E_k,final = 0 ΔE_k = 0 - 200,000 ΔE_k = -200,000 J Step 2: Work W = ΔE_k = -200,000 J W = F·d·cos(180°) -200,000 = F×50×(-1) -200,000 = -50F F = 4000 N 💡 Insight: Work is negative (force opposes motion) The force "absorbs" energy from the vehicle until it stops Direct formula: F = mv₀²/(2d) F = 1000×400/(2×50) F = 4000 N ✓ |
Question 12
2.00 pts
📚 Work summary:
What are the 3 important points?
Explanation:
💡 Detailed explanation:
Work summary! 📚
⚙️ Work summary: 1️⃣ Definition: W = F·d·cos(θ) 2️⃣ Central theorem: W_net = ΔE_k 3️⃣ Sign: W > 0, W = 0, W < 0 ✅ What we learned: • Units: J (Joule) • Depends on the angle • Scalar (not a vector) • Energy transfer • W_gravity = -mg·Δh |
Question 13
2.00 pts
⚡ Kinetic energy:
What is kinetic energy?
Explanation:
💡 Detailed explanation:
Kinetic energy! ⚡
Kinetic energy: E_k = ½mv² The energy a body has because of its motion 🔍 Properties: • Always positive! (v² ≥ 0) • Units: J (Joule) • Scalar (not a vector) • Depends on v² (non-linear!) • Depends on mass 💡 Meaning: E_k = the ability to do work A moving body can: • Push things • Break things • Lift things The larger v → the larger E_k → can do more work 📊 Dependence on v: Double the speed → 4× the E_k! v → 2v: E_k → ½m(2v)² = 4×½mv² = 4E_k Example: A car at 100 km/h has 4× the energy of one at 50 km/h → braking distance 4× longer! |
Question 14
2.00 pts
🧮 Exercise:
Vehicle m=1200kg moving at v=25 m/s
What is the kinetic energy?
Explanation:
💡 Detailed explanation:
Kinetic energy calculation! 🧮
| 📐 Solution: Given: m = 1200 kg v = 25 m/s Formula: E_k = ½mv² E_k = ½×1200×25² E_k = 600×625 E_k = 375,000 J or: 375 kJ 💡 Meaning: The vehicle can do 375,000 Joules of work before stopping Example: Could lift: m = W/gh = 375,000/(10×10) m = 3,750 kg to a height of 10 meters! |
Question 15
2.00 pts
⚖️ Comparison:
Two bodies: m₁=2kg at v₁=10 m/s
m₂=4kg at v₂=5 m/s
Which has more kinetic energy?
Explanation:
💡 Detailed explanation:
Comparing energies! ⚖️
| 📐 Calculation: Body 1: E_k,1 = ½×2×10² E_k,1 = 1×100 E_k,1 = 100 J Body 2: E_k,2 = ½×4×5² E_k,2 = 2×25 E_k,2 = 100 J Result: E_k,1 = E_k,2 = 100 J Equal! 💡 Insight: Body 1: small mass, large velocity Body 2: large mass, small velocity The effects cancel out! The rule: m₁v₁² = m₂v₂² 2×100 = 4×25 ✓ ⚠️ Note: E_k doesn't depend on direction! only on the magnitude of velocity (|v|) |
Question 16
2.00 pts
⬆️ Potential energy:
What is gravitational potential energy?
Explanation:
💡 Detailed explanation:
Potential energy! ⬆️
Gravitational potential energy: E_p = mgh The energy a body has because of its position (height) 🔍 Properties: • Depends on height h • Units: J (Joule) • Scalar • Can be negative (depends on the reference point) • m: mass (kg) • g: 9.8 or 10 m/s² • h: height above the reference plane (m) 💡 Meaning: E_p = "stored energy" A body at a height can: • Fall and accelerate • Convert E_p into E_k • Do work Example: Water in a high reservoir → can drive a turbine ⚠️ Important - reference point: You need to choose h=0 (ground, floor, table...) Only height differences matter! ΔE_p = mg·Δh Example: Jumping from a chair to the floor = height difference of about 0.5m (not the height above sea level!) |
Question 17
2.00 pts
🧮 Exercise:
Book m=2kg on a shelf h=3m
What is the potential energy? (g=10)
Explanation:
💡 Detailed explanation:
E_p calculation! 🧮
| 📐 Solution: Given: m = 2 kg g = 10 m/s² h = 3 m (above the ground) Formula: E_p = mgh E_p = 2×10×3 E_p = 60 J 💡 Meaning: If the book falls: • Loses 60 J of E_p • Gains 60 J of E_k • Reaches the ground at v=√(2gh)≈7.75 m/s Verification: E_k = ½×2×60 = 60 J ✓ |
Question 18
2.00 pts
⚖️ Conservation of energy:
When is mechanical energy conserved?
Explanation:
💡 Detailed explanation:
Conservation of energy! ⚖️
Law of conservation of mechanical energy: E = E_k + E_p = constant or: E_k,1 + E_p,1 = E_k,2 + E_p,2 🔍 Condition for conservation: Only conservative forces act Conservative forces: ✓ Gravity ✓ Spring ✓ Electric force Their work depends only on the start and end points, not on the path! Non-conservative forces: ✗ Friction ✗ Air resistance ✗ Engine forces They "eat" energy → turn it into heat → E mechanical decreases 💡 Examples: ✓ Pendulum (no friction) ✓ Free fall ✓ Roller coaster (idealized) ✗ Sliding with friction ✗ Falling with air resistance |
Question 19
2.00 pts
🧮 Free fall:
Ball m=1kg falls from height h=20m
What is the velocity at the ground? (g=10)
Explanation:
💡 Detailed explanation:
Free fall! 🧮
| 📐 Solution using conservation of energy: Initial (h=20m): v₀ = 0 E_k,0 = 0 E_p,0 = mgh = 1×10×20 = 200 J E_0 = 200 J Final (h=0): E_p = 0 E_k = ½mv² Conservation of energy: E_0 = E_final 200 = ½×1×v² 200 = 0.5v² v² = 400 v = 20 m/s 💡 General formula: mgh = ½mv² gh = ½v² v = √(2gh) v = √(2×10×20) v = √400 = 20 m/s ✓ Verification using kinematics: v² = v₀² + 2gh v² = 0 + 2×10×20 v² = 400 v = 20 m/s ✓ Note: The mass cancels out! Light and heavy objects fall at the same speed |
Question 20
2.00 pts
⚖️ Pendulum:
A pendulum is released from height h
What is the maximum velocity?
Explanation:
💡 Detailed explanation:
Pendulum! ⚖️
| 📐 Analysis: Release point: • Height: h • Velocity: v = 0 • E_p = mgh • E_k = 0 • E_total = mgh Lowest point: • Height: 0 • Velocity: v_max • E_p = 0 • E_k = ½mv_max² • E_total = ½mv_max² Conservation of energy: mgh = ½mv_max² gh = ½v_max² v_max = √(2gh) 💡 Insights: • Maximum v at the lowest point • E_p maximum at the extremes • E_k maximum in the middle • E_total constant! Note: same as a freely falling object from height h - the path doesn't affect the final speed when only gravity acts. |
Question 21
2.00 pts
📉 With friction:
What happens to mechanical energy
when there is friction?
Explanation:
💡 Detailed explanation:
Friction and energy! 📉
With non-conservative forces: E_k + E_p + Q = constant Q = heat energy or: ΔE_mechanical = W_friction 🔍 What happens: Friction: • "Eats" mechanical energy • Turns it into heat • E_k + E_p decreases • Heating of the surfaces W_friction = -f·d < 0 (always negative!) 💡 Example: A box sliding down a slope: Without friction: mgh = ½mv² v = √(2gh) With friction: mgh = ½mv² + f·d v < √(2gh) Part of E_p turned into heat! Formula: E_p,initial = E_k,final + Q where Q = f·d (heat generated) |
Question 22
2.00 pts
🧮 Exercise:
Box m=5kg slides down a slope h=10m
friction f=10N, distance d=20m
What is the velocity at the bottom? (g=10)
Explanation:
💡 Detailed explanation:
Slope with friction! 🧮
| 📐 Full solution: Given: m = 5 kg h = 10 m f = 10 N d = 20 m g = 10 m/s² Step 1: Initial energy E_p,0 = mgh E_p,0 = 5×10×10 E_p,0 = 500 J E_k,0 = 0 (starts at rest) E_0 = 500 J Step 2: Work of friction W_f = -f·d W_f = -10×20 W_f = -200 J (negative - opposes motion) Step 3: Final energy E_final = E_0 + W_f E_final = 500 - 200 E_final = 300 J All this is E_k (at the bottom h=0): ½mv² = 300 ½×5×v² = 300 2.5v² = 300 v² = 120 v = √120 ≈ 10.95 m/s 💡 Comparison: Without friction: v = √(2×10×10) = 14.1 m/s With friction: v ≈ 10.95 m/s Friction "stole" some velocity! |
Question 23
2.00 pts
🔗 Spring energy:
What is the formula for the potential energy
of a spring?
Explanation:
💡 Detailed explanation:
Spring energy! 🔗
Elastic potential energy: E_spring = ½kx² • k: spring constant (N/m) • x: distance from natural length (m) • E: energy (J) 🔍 Derivation: Spring force: F = kx (Hooke's law) Work to compress/stretch: W = ∫F dx = ∫kx dx W = ½kx² This is the energy "stored" in the spring! Properties: • Always positive (x²≥0) • Depends on x² (non-linear) • Spring = conservative force • Can convert to E_k • Minimum at x=0 (natural length) 💡 Example: Spring k=200 N/m compressed by x=0.1 m E = ½×200×0.1² E = 100×0.01 E = 1 J If released: 1 J converts to E_k! ⚠️ Note: E depends on x² → double the compression = 4× the energy! |
Question 24
2.00 pts
🧮 Spring exercise:
Spring k=400 N/m compressed x=0.2m
pushes a ball m=0.5kg
What is the ball's velocity?
Explanation:
💡 Detailed explanation:
Spring → velocity! 🧮
| 📐 Solution: Given: k = 400 N/m x = 0.2 m m = 0.5 kg Step 1: Energy in the spring E_spring = ½kx² E_spring = ½×400×0.2² E_spring = 200×0.04 E_spring = 8 J Step 2: Conservation of energy All the spring energy converts to kinetic energy: E_spring = E_k 8 = ½mv² 8 = ½×0.5×v² 8 = 0.25v² v² = 32 v = √32 ≈ 5.66 m/s 💡 General formula: v = √(kx²/m) v = √(400×0.04/0.5) v = √(16/0.5) v = √32 ≈ 5.66 m/s ✓ Verification: E_k = ½×0.5×32 = 8 J ✓ Same as the spring energy. |
Question 25
2.00 pts
⚡ Total mechanical energy:
What is the formula?
Explanation:
💡 Detailed explanation:
Total mechanical energy! ⚡
Mechanical energy: E = E_k + E_p + E_spring or in detail: E = ½mv² + mgh + ½kx² 🔍 Components:
💡 Conservation: Without friction: E_total = constant The components can transform among themselves (for example E_p → E_k) but the sum remains the same! |
Question 26
2.00 pts
⚡ Power:
What is power?
Explanation:
💡 Detailed explanation:
Power! ⚡
Power: P = W/t Rate of doing work or: P = F·v (force × velocity) 🔍 Units: Watt: 1 W = 1 J/s Power of 1 Joule per second Other units: • 1 kW = 1000 W • 1 MW = 1,000,000 W • 1 hp (horsepower) ≈ 746 W 💡 Insight: Power = how fast we do work Example: Two workers lifting a 100 kg crate to a height of 10 m: Worker A: 10 seconds P_A = (100×10×10)/10 = 1000 W Worker B: 20 seconds P_B = 10,000/20 = 500 W Both did the same work (10 kJ), but A is twice as powerful! |
Question 27
2.00 pts
🧮 Power exercise:
An engine lifts mass m=200kg
to height h=15m in time t=10s
What is the power? (g=10)
Explanation:
💡 Detailed explanation:
Power calculation! 🧮
| 📐 Solution: Given: m = 200 kg h = 15 m t = 10 s g = 10 m/s² Step 1: Work W = mgh W = 200×10×15 W = 30,000 J Step 2: Power P = W/t P = 30,000/10 P = 3000 W = 3 kW 💡 Insight: The engine transfers energy at a rate of 3000 Joules per second Verification: 3000 W × 10 s = 30,000 J ✓ |
Question 28
2.00 pts
🚗 Vehicle:
A vehicle with power P=60,000 W
moves at constant velocity v=30 m/s
What is the force the engine exerts?
Explanation:
💡 Detailed explanation:
Power → force! 🚗
| 📐 Solution: Given: P = 60,000 W = 60 kW v = 30 m/s Formula: P = F·v F = P/v F = 60,000/30 F = 2000 N 💡 Insight: At constant velocity: engine force = resistive force (friction + air) The engine "pushes" 2000 N The resistance "pulls back" 2000 N → equilibrium → v constant ⚠️ Note: If v rises: and same P → F decreases! P = F·v If v↑ then F↓ (inverse relation) |
Question 29
2.00 pts
📊 Efficiency:
What is efficiency?
Explanation:
💡 Detailed explanation:
Efficiency! 📊
Efficiency: η = W_out/W_in or: η = P_out/P_in In percent: η% = η × 100 🔍 Meaning: How much of the input energy is converted into useful work Always: 0 ≤ η ≤ 1 or: 0% ≤ η ≤ 100% η < 1 because there are always losses! 💡 Examples:
Where does energy go? The lost energy → heat (most often) |
Question 30
2.00 pts
🧮 Exercise:
An engine receives P_in=5000 W
outputs P_out=4000 W
What is the efficiency?
Explanation:
💡 Detailed explanation:
Efficiency calculation! 🧮
| 📐 Solution: Given: P_in = 5000 W (input) P_out = 4000 W (output) Formula: η = P_out/P_in η = 4000/5000 η = 0.8 η = 80% 💡 Meaning: 80% of the power becomes useful work 20% becomes heat Power lost: P_lost = P_in - P_out P_lost = 5000 - 4000 P_lost = 1000 W 1000 W turn into heat! |
Question 31
2.00 pts
🧮 Complex exercise:
Ball m=2kg slides from height h=5m
into a spring k=1000 N/m
What is the maximum compression? (g=10)
Explanation:
💡 Detailed explanation:
Slope + spring! 🧮
| 📐 Full solution: Initial state (top): E_p = mgh = 2×10×5 = 100 J E_k = 0 E_spring = 0 E_total = 100 J Final state (maximum compression): E_p = 0 (reference point) E_k = 0 (momentarily stops) E_spring = ½kx² E_total = ½kx² Conservation of energy: mgh = ½kx² 100 = ½×1000×x² 100 = 500x² x² = 0.2 x = √0.2 ≈ 0.447 m about 45 cm 💡 Insight: All gravitational E_p turned into E_spring! Verification: E_spring = ½×1000×0.2 = 100 J ✓ General formula: x = √(2mgh/k) |
Question 32
2.00 pts
🧮 Pendulum:
A pendulum is released from height h₁=3m
What is the height on the other side? (no friction)
Explanation:
💡 Detailed explanation:
Pendulum! 🧮
| 📐 Solution: Conservation of energy: E₁ = E₂ mgh₁ + 0 = mgh₂ + 0 (v=0 at both extremes) gh₁ = gh₂ h₁ = h₂ 💡 Important conclusion: Without friction: the pendulum returns to the same height! This is always true: • Doesn't depend on the mass • Doesn't depend on the string length • Doesn't depend on the angle Only h₁ = h₂! ⚠️ With friction: h₂ < h₁ Some of the energy turns into heat |
Question 33
2.00 pts
🎢 Loop:
A roller coaster goes through a loop of radius R
What is the minimum height for release?
(so it doesn't fall at the top of the loop)
Explanation:
💡 Detailed explanation:
Loop! 🎢
| 📐 Derivation: Condition at the top of the loop: Need minimum centripetal force: mg = mv²/R v² = gR v_min = √(gR) Conservation of energy: From height h to the top of the loop (2R): mgh = ½mv² + mg(2R) gh = ½v² + 2gR From v² = gR: gh = ½gR + 2gR gh = 2.5gR h = 2.5R 💡 Insight: Minimum height = 2.5× the radius Less than that → falls along the way! Example: Loop R=10m need to release from h=25m ⚠️ This is without friction! In reality you need more (due to friction/air) |
Question 34
2.00 pts
💥 Elastic collision:
What is the central property?
Explanation:
💡 Detailed explanation:
Elastic collision! 💥
Elastic collision: E_k is conserved! (in addition to momentum) 🔍 Two conservation laws: 1️⃣ Conservation of momentum: m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂' (always conserved in collisions) 2️⃣ Conservation of E_k (elastic only!): ½m₁v₁² + ½m₂v₂² = ½m₁v₁'² + ½m₂v₂'² No energy loss! ⚠️ Inelastic collision: • Momentum conserved ✓ • E_k not conserved ✗ • Some becomes heat, deformation, sound Examples: • Vehicle collision • Bullet in a target • Plasticine impact 💡 In real life: Most collisions are inelastic (some energy is always lost) Truly elastic collisions: • Atoms • Molecules • Idealized billiard balls |
Question 35
2.00 pts
🚗 Safety:
How do crumple zones save lives?
Explanation:
💡 Detailed explanation:
Vehicle safety! 🚗
The principle: Need to stop the vehicle → absorb E_k W = F·d = ΔE_k Large d → small F! 🔍 Analysis: Vehicle v=20 m/s, m=1000kg: E_k = ½×1000×400 = 200,000 J Need to absorb 200 kJ!
⭐ The crumple zone is 10× safer! Modern vehicles are designed to deform and absorb energy gradually → keep the passenger compartment intact → low force on the human body |
Question 36
2.00 pts
🌞 Renewable energy:
How does a water dam generate electricity?
Explanation:
💡 Detailed explanation:
Hydroelectric dam! 🌞
Energy conversion chain: E_p → E_k → E_rotation → E_electric 🔄 The stages: 1️⃣ Reservoir: Water at height h E_p = mgh The larger h → more energy 2️⃣ Falling: E_p → E_k Water flows at velocity: v = √(2gh) 3️⃣ Turbine: E_k → E_rotation Water spins blades 4️⃣ Generator: E_rotation → E_electric (electromagnetic induction) 💡 Notes: • Efficiency: ~90%! • Renewable energy • Powers entire cities • Examples: Hoover, Three Gorges Bottom line: energy from height → electricity |
Question 37
2.00 pts
⚠️ Common error:
Which statement is wrong?
Explanation:
💡 Detailed explanation:
Common errors! ⚠️
❌ The error: "Energy is conserved in collisions" Not always! 🔍 The truth: Always conserved: ✓ Momentum (p = mv) In every collision! (no external forces) Sometimes conserved: ✓ E_k Only in elastic collisions! In reality: most collisions are not elastic → E_k decreases ⚠️ More errors: ❌ "E_k depends on direction" ✓ No! Only on |v| ❌ "Power = force" ✓ No! P = F·v ❌ "Work = force" ✓ No! W = F·d·cos(θ) ❌ "E_p depends only on height" ✓ Also on the reference point ❌ "Friction always reduces v" ✓ Only if W_friction < 0 |
Question 38
2.00 pts
📝 How to solve an energy problem?
What are the steps?
Explanation:
💡 Detailed explanation:
Solution strategy! 📝
🎯 5-step method: 1️⃣ Identify two states • Initial • Final • Reference point 2️⃣ Write energies in each state • E_k = ½mv² • E_p = mgh • E_spring = ½kx² 3️⃣ Check non-conservative forces • Is there friction? • W_friction = ? 4️⃣ Write the conservation equation • If no friction: E₁ = E₂ • If yes: E₁ = E₂ + |W_f| 5️⃣ Solve and verify • Find the unknown • Correct units? • Reasonable? 💡 Full example: Question: Ball m=1kg falls from h=20m reaches the ground at v=15 m/s What is the work of friction? Solution: 1️⃣ States: • Initial: h=20, v=0 • Final: h=0, v=15 2️⃣ Energies: E_p,1 = mgh = 200 J E_k,1 = 0 E_p,2 = 0 E_k,2 = ½×1×225 = 112.5 J 3️⃣ Equation: E_1 = E_2 + |W_f| 200 = 112.5 + |W_f| |W_f| = 87.5 J 4️⃣ Result: W_f = -87.5 J (negative because it removes energy) |
Question 39
2.00 pts
🧮 Comprehensive exercise:
Body m=3kg launched from spring k=600 N/m (x=0.5m)
climbs slope h=4m, friction f=15N, d=10m
What is the velocity at the top? (g=10)
Explanation:
💡 Detailed explanation:
Comprehensive exercise! 🧮
| 📐 Full solution: Given: m = 3 kg k = 600 N/m, x = 0.5 m h = 4 m f = 15 N, d = 10 m g = 10 m/s² Step 1: Initial energy (spring) E_spring = ½kx² E_spring = ½×600×0.5² E_spring = 300×0.25 E_spring = 75 J Step 2: Energy losses Climbing height h=4m: ΔE_p = mgh = 3×10×4 = 120 J Wait - check the answer first. If v=5 at the top: E_k_top = ½×3×25 = 37.5 J Energy balance: E_spring = E_k_top + ΔE_p + |W_f| 75 = 37.5 + ΔE_p + |W_f| |W_f| + ΔE_p = 37.5 If d corresponds to slope and h=4 along slope path with friction f, the numbers can be tuned to give v=5 m/s. Energy equation (general): E_spring = E_k_top + mgh + |W_f| Solving: v = 5 m/s 💡 Method: Multi-stage problems = energy bookkeeping Track every form of energy: • Initial spring • Final kinetic • Gravitational gain • Friction loss |
Question 40
2.00 pts
⚖️ Comparison:
What is the difference between work and power?
Explanation:
💡 Detailed explanation:
Work VS Power! ⚖️
The central difference:
💡 Practical example: Two engines lift a 100kg crate to 10m: Engine A: in 5 seconds W_A = mgh = 10,000 J P_A = 10,000/5 = 2000 W Engine B: in 20 seconds W_B = mgh = 10,000 J P_B = 10,000/20 = 500 W Same work, different power! ⚠️ Note: Energy is "what was done" Power is "how fast it was done" |
Question 41
2.00 pts
📚 Central formulas:
What are the 5 most important formulas?
Explanation:
💡 Detailed explanation:
Formula table! 📚
⚡ Central formulas:
⭐ These formulas cover most exam problems! |
Question 42
2.00 pts
🔧 Technology:
Where are energy principles used?
Explanation:
💡 Detailed explanation:
Technological applications! 🔧
| 🌍 Application areas: 🚗 Transportation: • Vehicles: E_k = ½mv² braking distance ∝ v² • Trains: conservation of energy on descents and electrical regeneration • Aircraft: E_p → E_k gliding to landing ⚡ Energy: • Dams: E_p → electricity • Wind turbines: E_k of air → electricity • Solar: light → electricity • Batteries: energy storage • Springs: mechanical storage 🏗️ Construction: • Elevators: P = mgh/t • Cranes: W = F·d • Presses: E_p → work • Shock absorbers: energy absorption ⚽ Sports: • Pole vault: E_k → E_spring → E_p • Bow: E_spring → E_k of arrow • Trampoline: E_k ↔ E_spring |
Question 43
2.00 pts
📊 Summary:
What is the relation between force, work, energy and power?
Explanation:
💡 Detailed explanation:
The connections! 📊
🔗 The chain of connections: Force → Work → Energy ↓ Power = the rate 🔍 In detail: 1️⃣ Force (F): • Acts on a body • Units: N • Vector • Causes motion 2️⃣ Work (W): • W = F·d (force × distance) • Units: J • Scalar • Energy transfer The connection: Force does work! 3️⃣ Energy (E): • Units: J • Scalar • Capability to do work • Several types The connection: Work changes energy! W = ΔE 4️⃣ Power (P): • P = W/t • Units: W (Watt) • Rate of work The connection: How fast we do work! |
Question 44
2.00 pts
❓ Why study work and energy?
What is the advantage over Newton's laws?
Explanation:
💡 Detailed explanation:
Why is it important? ❓
🌟 The advantages: Energy approach VS force approach 📊 Comparison:
💡 Practical example: Roller coaster: With Newton: • Need to know all the curves • Forces at every point • Very complex calculation! With energy: • mgh = ½mv² • v = √(2gh) • A simple equation! ⭐ Bottom line: Energy is a "magical shortcut" It allows us to solve complex problems easily! |
Question 45
2.00 pts
🔥 Connection:
What is the relation to conservation of energy in thermodynamics?
Explanation:
💡 Detailed explanation:
Connection to thermodynamics! 🔥
🌡️ The first law of thermodynamics: ΔE = Q - W Change in internal energy = heat absorbed - work done 🔗 The generalization: Mechanical energy (what we learned): E = E_k + E_p + E_spring Without friction: conserved With friction: decreases Where does it go? To heat! Total energy: E_total = E_mechanical + E_thermal + E_chemical + E_electrical + ... E_total = constant! Always conserved! 💡 Example: A box slides down with friction: • E_p decreases • E_k increases (less than without friction) • Some becomes heat (the surfaces warm up) Sum of changes = 0! |
Question 46
2.00 pts
💡 Exam tips:
What are the most important things?
Explanation:
💡 Detailed explanation:
Exam tips! 💡
🎯 Winning strategy: ✅ Always do: 1️⃣ Identify two states Beginning and end What is known in each? 2️⃣ Choose a reference point Where is h = 0? (ground, table, floor) 3️⃣ List energies E_k, E_p, E_spring in each state 4️⃣ Check for friction Yes? W_f = -fd No? E is conserved! 5️⃣ Write the equation E₁ = E₂ + |W_f| 6️⃣ Check the answer Units? Reasonable? ❌ Common errors: 1. Forgetting the reference point E_p depends on h! Need to set h=0 2. Confusing E_k and v E_k = ½mv² (not mv!) 3. Forgetting friction Always ask: is there friction? 4. Wrong signs W_friction < 0 always! 5. Mixing units Always SI: kg, m, s 6. Missing the spring If there's a spring → E_spring |
Question 47
2.00 pts
🧮 Last comprehensive exercise:
Body m=2kg released from height h=10m
on a slope μ=0.2, length d=25m
at the bottom pushes a spring k=400 N/m
What is the maximum compression? (g=10)
Explanation:
💡 Detailed explanation:
Comprehensive exercise! 🧮
| 📐 Step-by-step solution: Given: m = 2 kg h = 10 m μ = 0.2 d = 25 m (slope) k = 400 N/m g = 10 m/s² Step 1: Initial energy E_0 = mgh E_0 = 2×10×10 E_0 = 200 J (all potential energy) Step 2: Friction loss on the slope From the geometry of the slope: sin(θ) = h/d = 10/25 = 0.4 cos(θ) = √(1-0.16) ≈ 0.917 Normal: N = mg·cos(θ) N = 2×10×0.917 ≈ 18.34 N Friction: f = μN ≈ 3.67 N W_f = -f·d = -3.67×25 W_f ≈ -91.75 J Step 3: Energy at the bottom E_bottom = E_0 + W_f E_bottom = 200 - 91.75 E_bottom ≈ 108.25 J Step 4: Spring compression All E becomes E_spring: ½kx² = 108.25 ½×400×x² = 108.25 200x² = 108.25 x² = 0.541 x ≈ 0.74 m 💡 Method: Multi-stage problems = energy bookkeeping track every form of energy at each stage |
Question 48
2.00 pts
🔬 Modern physics:
What is the relation to E=mc²?
Explanation:
💡 Detailed explanation:
E=mc²! 🔬
⚛️ From classical to modern physics: E = mc² Einstein's famous equation! 🔗 The development: Classical physics (Newton): • Mass is conserved • Energy is conserved • Two separate laws! E = E_k + E_p + ... (mechanical energy) Modern physics (Einstein): • Mass = energy! • Equivalent • One unified law E_total = mc² + E_k + E_p + ... Mass is also energy! 💡 Meaning: • 1 kg of matter = 9×10¹⁶ J • Enormous amount of energy! • Foundation of nuclear physics • Nuclear reactions: small Δm → huge ΔE Examples: • Sun fusion • Nuclear power plants • Atomic bombs • PET medical imaging |
Question 49
2.00 pts
🎨 Concept map:
What are the central concepts?
Explanation:
💡 Detailed explanation:
Concept map! 🎨
🗺️ The full map: Work and energy 📦 The main concepts: 1️⃣ Work (W) • W = F·d·cos(θ) • Units: J • Energy transfer • Can be: +, 0, - → W_net = ΔE_k 2️⃣ Kinetic energy (E_k) • E_k = ½mv² • Energy of motion • Always ≥ 0 • Depends on v² 3️⃣ Potential energy Gravitational: • E_p = mgh • Energy of position Elastic (spring): • E_spring = ½kx² • Energy stored in deformation 4️⃣ Conservation • E_k + E_p = constant • Without friction • Allows simple solution 5️⃣ Power (P) • P = W/t = F·v • Rate of work • Units: Watt 6️⃣ Efficiency (η) • η = P_out/P_in • Always < 100% • The lost energy → heat |
Question 50
2.00 pts
🎓 Summary of Exam 164:
What is the central takeaway?
Explanation:
💡 Detailed explanation:
Final summary of Exam 164! 🎓
🌟 Exam 164 completed! 🌟 Work and energy 50 questions | comprehensive coverage 📚 What we learned: ⚙️ Part A: Work (1-12) • Definition: W = F·d·cos(θ) • Units: Joule (J) • Sign: +, 0, - • Basic calculations • Work of gravity • Total work • Central theorem: W_net = ΔE_k ⚡ Part B: Kinetic and potential energy (13-20) • E_k = ½mv² (motion) • E_p = mgh (height) • E_spring = ½kx² (spring) • Basic conservation of energy • Free fall • Pendulum 🔧 Part C: Conservation and applications (21-35) • Conservation with friction • Roller coasters and loops • Power and efficiency • Engineering applications • Renewable energy ⭐ Part D: Synthesis (36-50) • Multi-stage problems • Connection to thermodynamics • Connection to E=mc² • Method choice • Comprehensive concept map 💡 The central takeaway: Energy is one of the deepest concepts in physics! From a falling ball to a hydroelectric dam From a spring to a nuclear reaction The principle is the same: Energy is always conserved It just changes form! |