23 worked answers · short steps first, explanations when you need them
Chapter 5 Exercises
Textbook pages 88–90
Question 1
In each case, say whether the stated force does positive, negative or zero work: (a) a person lifts a bucket, (b) gravity acts on the rising bucket, (c) friction acts on a sliding body, (d) an applied force keeps a body moving at constant speed on a rough floor, (e) air resistance slows a pendulum.
Steps only
Work is positive when force points with displacement, negative when it points against it, and zero when the two are perpendicular or displacement is zero.
A 2 kg body starts from rest on a horizontal table. A 7 N force moves it for 10 s; the coefficient of kinetic friction is 0.1. Find (a) work by the applied force, (b) work by friction, (c) net work and (d) change in kinetic energy. Use g=9.8 m/s².
Steps only
Friction = μmg = 0.1×2×9.8 = 1.96 N. Net force = 7−1.96 = 5.04 N, so a=5.04/2=2.52 m/s².
Answer
(a) 882 J; (b) −247 J; (c) about 635 J; (d) about 635 J.
Question 3
For each of the four potential-energy graphs in Fig. 5.11, identify where the particle cannot be for the marked total energy, state the minimum possible total energy, and suggest a physical example.
Steps only
Kinetic energy is K=E−V(x). A particle cannot be where V(x)>E, because that would make K negative.
Answer
A particle cannot be where V>E.
(i) x>a, minimum E=0;
(ii) no allowed region at the marked E;
(iii) between a and b, minimum E=−V₁;
(iv) forbidden in the two indicated barrier intervals, minimum E=−V₁.
Question 4
For V(x)=kx²/2 with k=0.5 N/m, show that a particle with total energy 1 J turns back at x=±2 m.
Steps only
At a turning point, the particle momentarily stops, so K=0 and E=V.
Answer
The turning points are x=−2 m and x=+2 m.
Question 5
Answer: (a) Is the heat that burns a rocket casing supplied by the rocket or atmosphere? (b) Why is gravitational work on a comet zero over one complete orbit? (c) Why can a satellite speed up as it moves closer to Earth despite atmospheric resistance? (d) A person walks 2 m holding a 15 kg load, then walks 2 m pulling a rope that raises a 15 kg mass. In which case is more work done on the load?
Steps only
(a) The rocket’s mechanical energy is converted to heat by air resistance, so the rocket loses energy. (b) Gravity is conservative; its work over a closed path is zero.
Answer
(a) Rocket; (b) gravity is conservative; (c) falling potential energy can increase speed despite some energy lost to drag; (d) pulling the rope does more work on the load.
Question 6
Choose the correct word: (a) Positive work by a conservative force makes potential energy increase/decrease/remain unchanged. (b) Work against friction loses kinetic/potential energy. (c) Change of total momentum of a many-particle system depends on external force/internal forces. (d) In an inelastic collision, which stays unchanged: total kinetic energy, total linear momentum, or total energy of the two bodies?
Steps only
A conservative force doing positive work lowers potential energy. Friction removes kinetic energy. Internal forces cancel in the system total; external force changes total momentum.
Answer
(a) Decreases; (b) kinetic energy; (c) external force; (d) total linear momentum.
Question 7
Say whether each statement is true or false and give a reason: (a) in an elastic collision, each body's momentum and energy are conserved; (b) a system's total energy is always conserved despite external forces; (c) every force does zero work over a closed path; (d) an inelastic collision always has less final kinetic energy than initial kinetic energy.
Steps only
(a) False: total momentum and total kinetic energy are conserved for the two-body system, but can transfer between bodies. (b) False: an external force can transfer energy into or out of the system.
Answer
(a) False; (b) False; (c) False; (d) True.
Question 8
Answer with reasons: (a) Is kinetic energy conserved at every instant while elastic billiard balls are touching? (b) Is their total momentum conserved during contact? (c) What about an inelastic collision? (d) If the interaction potential depends only on the distance between ball centres, is the collision elastic or inelastic?
Steps only
During contact, some kinetic energy is temporarily stored as interaction potential energy, so kinetic energy need not be constant at each instant. For an isolated pair, momentum is conserved.
Answer
(a) No; (b) yes; (c) kinetic energy is not conserved but momentum is; (d) elastic.
Question 9
A body starts from rest and moves in one dimension with constant acceleration. Its power at time t is proportional to which of t¹ᐟ², t, t³ᐟ² or t²?
Steps only
The force F=ma is constant, and from rest v=at, so v is proportional to t.
Answer
Power is proportional to t (option ii).
Question 10
A body moves in one direction under constant power. Its displacement in time t is proportional to which of t¹ᐟ², t, t³ᐟ² or t²?
Steps only
P=mv(dv/dt)=constant, so integrating gives v² proportional to t and v proportional to t¹ᐟ².
Answer
Displacement is proportional to t³ᐟ² (option iii).
Question 11
A body moves 4 m along the z-axis under the constant force F=−i+2j+3k N. The unit vectors i, j and k point along the x, y and z axes. Find the work done.
Steps only
Displacement is 4k m. Work is the dot product F·s.
Answer
12 J.
Question 12
An electron has kinetic energy 10 keV and a proton 100 keV. Which is faster, and what is the ratio of their speeds? Masses: electron 9.11×10⁻³¹ kg, proton 1.67×10⁻²⁷ kg; 1 eV=1.60×10⁻¹⁹ J.
Steps only
Use v=√(2K/m). The energies are 1.60×10⁻¹⁵ J for the electron and 1.60×10⁻¹⁴ J for the proton.
Answer
The electron is faster; its speed is about 13.54 times the proton’s.
Question 13
A water drop of radius 2 mm falls 500 m. It reaches terminal speed halfway down and reaches the ground at 10 m/s. Find gravity’s work in each half and air resistance’s work over the whole fall. Use water density 1000 kg/m³ and g=9.8 m/s².
Steps only
The drop’s mass is ρ(4πr³/3)≈3.35×10⁻⁵ kg. Each half is 250 m, so gravity does mgs≈0.082 J in each half.
Answer
Gravity: about 0.082 J in each half. Air resistance over the fall: about −0.162 J.
Question 14
A gas molecule hits a horizontal wall at 200 m/s, making 30° with the normal, then rebounds at the same speed. Is momentum conserved in the collision? Is it elastic or inelastic?
Steps only
The molecule’s momentum changes direction, so its momentum alone is not conserved; the wall receives an equal and opposite impulse. Momentum of the molecule-plus-wall system is conserved.
Answer
Molecule alone: momentum is not conserved; molecule plus wall: momentum is conserved. The collision is elastic.
Question 15
A pump fills a 30 m³ tank 40 m above the ground in 15 min. Pump efficiency is 30%. Find the electric power consumed. Use water density 1000 kg/m³ and g=9.8 m/s².
Steps only
Water mass=ρV=30,000 kg and time=900 s. Useful lifting power=mgh/t≈13,067 W.
Answer
The pump consumes about 43.6 kW of electric power.
Question 16
Two identical ball bearings are touching and at rest on a frictionless table. A third identical ball hits them head-on at speed V. For an elastic collision, which shown result is possible?
Steps only
Initially momentum is mV and kinetic energy is ½mV². Both must be conserved.
Answer
Case (ii): the incoming ball stops, and the far ball moves at speed V.
Question 17
A pendulum bob A released from 30° to the vertical hits an equal-mass bob B at rest. The collision is elastic. How high does A rise after the collision?
Steps only
In a head-on elastic collision between equal masses, the moving mass transfers its speed to the stationary mass.
Answer
Bob A does not rise after the collision (zero height).
Question 18
A pendulum bob is released from the horizontal. The pendulum length is 1.5 m. It loses 5% of its starting energy to air resistance. Find its speed at the lowest point. Use g=9.8 m/s².
Steps only
The bob falls through height 1.5 m. Its initial energy is mgL; 95% remains as kinetic energy at the bottom.
Answer
About 5.3 m/s.
Question 19
A 300 kg trolley carries a 25 kg sandbag and moves at 27 km/h on a frictionless track. Sand leaks vertically out at 0.05 kg/s. What is the trolley’s speed when the bag is empty?
Steps only
The sand leaves with the same horizontal speed as the trolley at the instant it leaks. There is no horizontal external force to change the system’s motion.
Answer
27 km/h (7.5 m/s).
Question 20
A 0.5 kg body moves in a straight line with v=ax³ᐟ², where a=5 m⁻¹ᐟ²s⁻¹. Find net work from x=0 to x=2 m.
Steps only
At x=0, u=0. At x=2, v=5×2³ᐟ²=10√2 m/s.
Answer
50 J.
Question 21
Wind of speed v flows perpendicular to a windmill’s swept area A for time t. Find (a) air mass passing through, (b) its kinetic energy, and (c) electrical power if 25% is converted, for A=30 m², v=36 km/h, air density 1.2 kg/m³.
Steps only
In time t the air volume is Avt, so its mass is ρAvt. Its kinetic energy is ½ρAv³t.
Answer
(a) ρAvt; (b) ½ρAv³t; (c) 4.5 kW.
Question 22
A person lifts a 10 kg mass through 0.5 m one thousand times. The lowering energy is dissipated. (a) Find work against gravity. (b) If fat provides 3.8×10⁷ J/kg and the body is 20% efficient, find fat used. Use g=9.8 m/s².
Steps only
Work against gravity=number of lifts×mgh=1000×10×9.8×0.5=49,000 J.
Answer
(a) 49 kJ; (b) about 6.45 g of fat.
Question 23
A family uses 8 kW. Sunlight supplies 200 W/m² on average, and 20% becomes useful electricity. (a) What solar-panel area is needed? (b) Compare it with a typical house roof.
Steps only
Each square metre supplies 0.20×200=40 W of useful power.
Answer
(a) 200 m²; (b) roughly the area of a 14 m × 14 m roof.