[CAP 2026 National] — 2026_CAP_Exam_with_solutions.pdf

Fonte: gare di altri paesi/Canada/all/2026_CAP_Exam_with_solutions.pdf · Apri PDF

Problema 1

Section A — Question 1

A block of wood was placed on an inclined plane and the angle of the plane was slowly increased until the block began to slide down. The angle remained constant from this point on. How would you describe the motion of the block?

A. The block moves at constant speed. B. The block accelerates. C. The block decelerates. D. The block moves for some distance and then stops.

Topic: Newtonian Mechanics Metodi: Free-Body Diagram, Kinematic Equations Competenze: Physical Reasoning Fonte: Testo (PDF) — p.5

Problema 2

Section A — Question 2

A strong electromagnet with an iron core is connected to a DC power source. A solid conducting ring placed on the core jumps upward when the power source is turned on. You repeat this experiment using several aluminum rings of the same diameter but different thicknesses. Which ring will jump the highest?

A. A 0.02-mm-thick ring at room temperature B. A 2-mm-thick ring at room temperature C. A 4-mm-thick ring at room temperature D. A 0.02-mm-thick ring cooled in liquid nitrogen E. A 2-mm-thick ring cooled in liquid nitrogen F. A 4-mm-thick ring cooled in liquid nitrogen

Topic: Electromagnetic Induction Metodi: Faraday’s Law of Induction, Lenz’s Law Competenze: Physical Reasoning Fonte: Testo (PDF) — p.5

Problema 3

Section A — Question 3

A heater is connected in the circuit shown below with the switch S open and the capacitor uncharged. The battery has no appreciable internal resistance. Which one of the following graphs best describes the power of the heater as a function of time, after closing the switch?

(The question refers to a circuit diagram and multiple graphs shown in the original PDF.)

Topic: Circuits Metodi: Kirchhoff’s Laws, Equivalent Circuit Reduction Competenze: Physical Reasoning, Diagrammatic Reasoning Fonte: Testo (PDF) — p.5

Problema 4

Section A — Question 4

The height on a flat mirror occupied by the image of a person’s face, of height $H$, is:

A. $H$ B. $H/2$ C. $H/4$ D. $H/8$ E. dependent on the distance between your face and the mirror.

Topic: Geometric Optics Metodi: Ray Tracing Competenze: Physical Reasoning Fonte: Testo (PDF) — p.6

Problema 5

Section A — Question 5

A parallel-plate capacitor has plate area $A$ and plate separation $d$. A dielectric of dielectric constant $\kappa$ fills half the gap (as shown in the figure in the original PDF). What is the capacitance $C$ of the capacitor?

A. $\frac{{\epsilon }_{0}A}{d}$

B. $\frac{\kappa {\epsilon }_{0}A}{d}$

C. $\frac{4\kappa {\epsilon }_{0}A}{d}$

D. $\frac{{\epsilon }_{0}A}{d}\cdot \frac{4\kappa }{3\kappa +1}$

E. $\frac{{\epsilon }_{0}A(3+2\kappa )}{d}$

F. $\frac{{\epsilon }_{0}A(3+2\kappa )}{5d}$

Topic: Electrostatics Metodi: Equivalent Circuit Reduction, Electric Potential Method Competenze: Physical Reasoning Fonte: Testo (PDF) — p.7

Problema 6

Section A — Question 6

The distance between a projectile and its launch point:

A. Always increases. B. First increases and then decreases. C. Always increases only if the launch angle is negative (below the horizontal). D. Can always increase for certain values of the initial velocity that are below the escape velocity. E. Can always increase for certain positive values of the launch angle. F. None of the above options is correct.

Topic: Newtonian Mechanics Metodi: Kinematic Equations, Vector Decomposition Competenze: Physical Reasoning, Mathematical Modeling Fonte: Testo (PDF) — p.7

Problema 7

Section A — Question 7

From the top of a tall building, you throw a $0.5\text{kg}$ ball straight up at an initial velocity of $5.0\text{m/s}$ and a $0.2\text{kg}$ ball straight down at an initial velocity of $0.5\text{m/s}$. Assume that both balls leave your hands from the same position at time $t=0\text{s}$ and ignore the effect of air resistance. Which of the following statements is true?

A. There will be a time $t>0\text{s}$ when the balls will be at the same position. B. There will be a time $t>0\text{s}$ when the balls will be moving at the same velocity. C. Both A and B are correct. D. None of the above options is correct.

Topic: Newtonian Mechanics Metodi: Kinematic Equations Competenze: Physical Reasoning Fonte: Testo (PDF) — p.7

Problema 8

Section A — Question 8

A $3.50\text{kg}$ mass is fastened to a vertical pole with two light strings, one above the other. When the system spins steadily about the pole’s axis, the strings stretch outward, as shown in the figure. If the tension in the upper string is measured to be $72\text{N}$, determine the speed of the mass in revolutions per minute.

A. 19 rpm B. 28 rpm C. 40 rpm D. 56 rpm E. 80 rpm

(The figure shows the geometry with string angles and distances; refer to PDF for diagram.)

Topic: Newtonian Mechanics Metodi: Free-Body Diagram, Vector Decomposition Competenze: Physical Reasoning, Mathematical Modeling Fonte: Testo (PDF) — p.8

Problema 9

Section A — Question 9

A steel spring of length $L$ and cross-sectional area $A$ is stretched by an amount $x$, storing potential energy $U$. The spring is then cut into half and the two shorter pieces are connected in parallel to form a new spring. By how much must this new spring be compressed so that it also stores potential energy $U$?

A. $x/2$ B. $x$ C. $\sqrt{2}x$ D. $2x$ E. $4x$

Topic: Elasticity & Materials, Conservation of Energy Metodi: Hooke’s Law, Energy Conservation Method Competenze: Physical Reasoning Fonte: Testo (PDF) — p.9

Problema 10

Section A — Question 10

The oven is at a temperature of $215°\text{C}$. You can put your hand inside the oven without an injury as long as you don’t touch anything. This is because:

A. The temperature of the air inside the oven is much lower than $215°\text{C}$. B. Thermal energy of the air inside the oven is much less than the thermal energy of the oven walls. C. The air has very low heat conductivity. D. Convection currents inside the oven move heat away from your hand instead of toward it.

Topic: Thermodynamics Metodi: Physical Modeling Competenze: Physical Reasoning Fonte: Testo (PDF) — p.9

Problema 11

Section A — Question 11

A metal object and a glass object have the same surface area and mass, and both are initially at room temperature. Metal has a lower specific heat and a higher heat conductivity than glass. If each object is dropped into an equal volume of $80°\text{C}$ water (insulated from the environment), we say that:

A. The metal object will reach equilibrium faster and end up at a higher temperature. B. The metal object will reach equilibrium faster and end up at a lower temperature. C. The metal object will reach equilibrium slower and end up at a higher temperature. D. The metal object will reach equilibrium slower and end up at a lower temperature. E. Both objects will end up at the same final temperature.

Topic: Thermodynamics Metodi: First Law of Thermodynamics, Physical Modeling Competenze: Physical Reasoning Fonte: Testo (PDF) — p.9

Problema 12

Section A — Question 12

A pendulum is suspended from the ceiling of an elevator and oscillates. If the elevator suddenly falls into free fall, how will the pendulum movement be affected?

A. The pendulum will remain stationary. B. No change will occur and the pendulum will continue its oscillation. C. The pendulum will perform uniform circular motion about the suspension point. D. Either option (A) or (C) may occur.

Topic: Oscillations & Waves, Newtonian Mechanics Metodi: Simple Harmonic Motion Analysis, Free-Body Diagram Competenze: Physical Reasoning Fonte: Testo (PDF) — p.10

Problema 13

Section A — Question 13

A body has an initial velocity of $8\text{m/s}$ directed north (from south to north). During the time interval $0\le t\le 4\text{s}$, the body is subjected to a constant force that gives it an acceleration of $3{\text{m/s}}^2$, directed toward the southwest (making an angle of $45°$ with the south–north axis). What is the approximate speed of the body at the instant $t=5\text{s}$?

A. $4\text{m/s}$ B. $8\text{m/s}$ C. $12\text{m/s}$ D. $16\text{m/s}$

Topic: Newtonian Mechanics Metodi: Kinematic Equations, Vector Decomposition Competenze: Mathematical Modeling, Physical Reasoning Fonte: Testo (PDF) — p.10

Problema 14

Section A — Question 14

A uniform rod is placed leaning against a vertical wall and a horizontal floor. In the following figures: smooth (frictionless) surfaces are shown with solid lines, and rough (frictional) surfaces are shown with dotted lines. We want the rod to remain in static equilibrium. Which of the following configurations can support the rod at rest?

(Refer to the three figures in the PDF showing different combinations of smooth/rough wall and floor.)

A. Only (3) B. Only (1) and (2) C. Only (2) and (3) D. Only (1) and (3) E. All the configurations can support the rod at rest.

Topic: Rigid Body Statics Metodi: Free-Body Diagram, Torque & Angular Momentum Analysis Competenze: Diagrammatic Reasoning, Physical Reasoning Fonte: Testo (PDF) — p.10

Problema 15

Section A — Question 15

Two plane mirrors are parallel and separated by a distance of $3.0\text{cm}$. As shown in the figure, the mirrors are parallel to the $x$-axis, equally spaced from it, and perpendicular to the $y$-axis. A point source $S$ and an observer $O$ both lie in the $xy$-plane with coordinates:

$x_s=-6.0\text{cm},y_s=-0.5\text{cm}$ $x_o=+2.0\text{cm},y_o=0.0\text{cm}$

How many images of the point source $S$ can the observer $O$ see in the mirrors?

A. 2 B. 4 C. 5 D. $\infty$

Topic: Geometric Optics Metodi: Ray Tracing, Superposition Principle Competenze: Diagrammatic Reasoning, Physical Reasoning Fonte: Testo (PDF) — p.11

Problema 16

Section A — Question 16

A particle of charge $q$ and mass $m$ moves at non-relativistic speed on a circular path under the influence of a uniform magnetic field $B$. As a result of this circular motion, the particle emits electromagnetic radiation with wavelength $\lambda$, that propagates at the speed of light, $c$. Which of the following options is the correct expression for $\lambda$?

A. $\lambda =\frac{qBc}{2\pi m}$

B. $\lambda =\frac{qB}{2\pi mc}$

C. $\lambda =\frac{2\pi mc}{qB}$

D. $\lambda =\frac{2\pi m}{qBc}$

Topic: Magnetism, Electromagnetism Metodi: Lorentz Force Analysis, Dimensional Analysis Competenze: Physical Reasoning, Unit Conversion Fonte: Testo (PDF) — p.12

Problema 17

Section A — Question 17

We pour $50{\text{cm}}^3$ of water onto the floor of a room with dimensions $5\text{m}\times 4\text{m}\times 3\text{m}$. After a long time, all the water evaporates. By approximately what factor does the average distance between two neighboring water molecules increase after evaporation?

A. 10 B. 100 C. 1000 D. 10,000

Topic: Kinetic Theory Metodi: Order-of-Magnitude Estimation, Kinetic Theory of Gases Competenze: Estimation & Approximation, Mathematical Modeling Fonte: Testo (PDF) — p.12

Problema 18

Section A — Question 18

A hollow tilted cylindrical vessel of negligible mass rests on a horizontal plane as shown. The diameter of the base is $a$ and the side of the cylinder makes an angle $\theta$ with the horizontal. Water is then slowly poured into the cylinder. The cylinder topples over when the water reaches a certain height $h$, given by:

A. $h=2a\tan \theta$ B. $h=a\tan 2\theta$ C. $h=a\tan \theta$ D. $h=\frac{a\tan \theta }{2}$

Topic: Fluid Mechanics, Rigid Body Statics Metodi: Hydrostatic Equilibrium, Torque & Angular Momentum Analysis Competenze: Physical Reasoning, Mathematical Modeling Fonte: Testo (PDF) — p.13

Problema 19

Section A — Question 19

A girl drops an apple from the window of a train which is moving on a straight track with speed increasing at a constant rate. The trajectory of the falling apple as seen by the girl is (ignore the effect of air drag):

A. parabolic and in the direction of the moving train B. parabolic and opposite to the direction of the moving train C. an inclined straight line pointing in the direction of the moving train D. an inclined straight line pointing opposite to the direction of the moving train

Topic: Newtonian Mechanics Metodi: Kinematic Equations, Vector Decomposition Competenze: Physical Reasoning Fonte: Testo (PDF) — p.13

Problema 20

Section A — Question 20

A hollow pendulum bob filled with water has a small hole at the bottom (as shown in the figure) through which water escapes at a constant rate. Which of the following statements describes the variation of the time period $T$ of the pendulum as the water flows out?

A. $T$ decreases at first and then increases. B. $T$ increases at first and then decreases. C. $T$ decreases throughout. D. $T$ is unaffected by the leakage of the water.

Topic: Oscillations & Waves, Newtonian Mechanics Metodi: Simple Harmonic Motion Analysis, Physical Modeling Competenze: Physical Reasoning Fonte: Testo (PDF) — p.14

Problema 21

Section A — Question 21

In the circuit shown below, the resistances are given in ohms and the battery is assumed ideal with emf equal to $3\text{V}$. The resistor that dissipates the most power is:

(Refer to the circuit diagram in the PDF showing resistors $R_1,R_2,R_3,R_4,R_5$.)

A. $R_1$ B. $R_2$ C. $R_3$ D. $R_4$ E. $R_5$

Topic: Circuits Metodi: Kirchhoff’s Laws, Equivalent Circuit Reduction Competenze: Physical Reasoning, Mathematical Modeling Fonte: Testo (PDF) — p.14

Problema 22

Section A — Question 22

The surface of a planet is found to be uniformly charged. When an object of mass $m$ and no charge is thrown upwards from the surface of the planet, it reaches a height $H$. An object of mass $m$ and charge $q$, thrown up with the same initial velocity, attains a height $H/2$. The height attained by an object of mass $m$ and charge $-q$ with the same initial conditions is:

A. $H$ B. $H/2$ C. $H/3$ D. $\infty$, since the particle never comes to rest.

Topic: Electrostatics, Gravitation Metodi: Conservation of Energy, Coulomb’s Law Competenze: Physical Reasoning, Mathematical Modeling Fonte: Testo (PDF) — p.15

Problema 23

Section A — Question 23

A planet of radius $R_p$ is revolving around a star of radius $R_{\star }$, which is at temperature $T_{\star }$. The distance between the star and the planet is $d$. The planet’s temperature is $f{T}_{\star }$. Note that the power radiated by a black body with surface area $A$ at temperature $T$ is proportional to $A{T}^4$. Assuming the planet and the star are perfect black bodies, $f$ is proportional to:

A. $\sqrt{R_{\star }/d}$ B. $R_{\star }/d$ C. $R_{\star }{R}_p/d^2$ D. $(R_{\star }/d{)}^4$

Topic: Astrophysics, Thermodynamics Metodi: Physical Modeling, Symmetry Argument Competenze: Mathematical Modeling, Physical Reasoning Fonte: Testo (PDF) — p.16

Problema 24

Section A — Question 24

A heater supplying constant power $P$ watts is switched on at time $t=0$ minutes to raise the temperature of a liquid kept in a calorimeter of negligible heat capacity. A student records the temperature of the liquid $T(t)$ in equal time intervals. A graph is plotted with $T(t)$ on the $y$-axis versus $t$ on the $x$-axis. Assume that there is no heat loss to the surrounding during heating and the liquid does not evaporate. Then:

A. the graph is always a straight line parallel to the time axis. B. the heat capacity of the liquid is inversely proportional to the slope of the graph. C. if some heat were lost at a constant rate to the surroundings during heating, the graph would be a straight line but with a larger slope. D. the internal energy of the liquid increases quadratically with time.

Topic: Thermodynamics Metodi: First Law of Thermodynamics, Graph Linearization Competenze: Physical Reasoning, Graph Linearization Fonte: Testo (PDF) — p.16

Problema 25

Section A — Question 25

We often observe that a rainbow appears as a circular arc in the sky. Which of the following best explains why a rainbow has this circular shape?

A. Because the Earth and its atmosphere are approximately spherical. B. Because the eye is spherical and perceives light in a symmetric way. C. Because each raindrop is spherical and disperses sunlight in a circularly symmetric pattern. D. Because the Sun is spherical and emits light symmetrically in all directions. E. Because raindrops polarize sunlight in a circular polarization mode.

Topic: Wave Optics, Geometric Optics Metodi: Ray Tracing, Symmetry Argument Competenze: Physical Reasoning Fonte: Testo (PDF) — p.16

Problema 26

Section B — Question 1: Far-flung fireflies

Fireflies produce light through a chemical process called bioluminescence. During the mating season, adult fireflies flash short pulses of greenish-yellow light with wavelengths centered around $562\text{nm}$. This light is often called “cold light” because the process generates almost no heat, making it nearly 100% efficient.

(a) A firefly consumes $141\mu \text{J}$ per second to produce a single flash. How many fireflies flashing simultaneously would be required to produce the same amount of visible light as a single yellow-green Christmas LED light? A typical yellow-green Christmas LED light uses $2\text{mW}$ of electrical power and has an efficiency of around 30%.

(b) Different species of fireflies have unique flashing patterns. Consider a species that produces single flashes. What is the maximum distance at which a person can detect this light from the firefly in the dark? Transmittance of light through the air is given as $T(r)=e^{-\alpha r}$, where $\alpha =2\times 10^{-4}{\text{m}}^{-1}$ can be used as the extinction coefficient of air. Assume that the person’s pupil dilates to $8\text{mm}$ in diameter in low-light conditions and that at least 6 photons reaching the eye almost simultaneously are required for this light to be detected by a human eye.

(c) Do you think this distance is realistic for a person observing a firefly on a dark night? Explain why or why not. In your answer, discuss the factors that would affect the actual detection distance.

Topic: Modern-Quantum Physics, Wave Optics Metodi: Photon Energy Relation, Physical Modeling, Order-of-Magnitude Estimation Competenze: Estimation & Approximation, Mathematical Modeling, Physical Reasoning Fonte: Testo (PDF) — p.17

Problema 27

Section B — Question 2: Against the wind!

A cyclist (total weight with the bike $75\text{kg}$) carefully measured the electric-motor power needed to ride without pedaling at different speeds on a horizontal road on a windless day. His observations are listed in Table 1.

Velocity [km/h]	Power [W]	Velocity [km/h]	Power [W]
0.5	5	16.0	189
1.0	10	18.0	220
1.5	15	20.0	254
2.0	20	22.0	290
4.0	41	24.0	330
6.0	63	26.0	374
8.0	85	28.0	421
10.0	108	30.0	473
12.0	133
14.0	160

1. Using these data, calculate the coefficient of rolling friction.
2. Determine the dependence of the drag force on the speed of the bike.
3. Calculate the drag coefficient for this cyclist.
4. What power is required to ride up a $30°$ slope at $30\text{km/h}$?
5. Estimate how much less power is required on the same $30°$ slope at $30\text{km/h}$ if the cyclist bends over the handlebars.

Notice:

• The drag force is proportional to the drag coefficient $C_d$, to the (approximate) front cross-sectional area of the cyclist ($A\approx 0.6{\text{m}}^2$), and to the air density ($\rho \approx 1.2{\text{kg/m}}^3$), and it depends on the velocity of air relative to the cyclist.
• The coefficient of rolling friction ${\mu }_r$ is the ratio of the rolling-resistance force to the normal force on the wheels and may be assumed independent of velocity.

Topic: Newtonian Mechanics, Fluid Mechanics Metodi: Experimental Data Analysis, Graph Linearization, Free-Body Diagram Competenze: Experimental Data Analysis, Graph Linearization, Mathematical Modeling Fonte: Testo (PDF) — p.18

Problema 28

Section B — Question 3: Physics of lightning

Tall cumulonimbus clouds (thunder-clouds) host strong temperature gradients: lower layers contain water droplets, while upper layers form ice crystals or graupel. Collisions between them generate static charge within the cloud which also polarises the ground, establishing an electric field from the cloud to the ground, much like in a parallel-plate capacitor. When the downward electric field beneath the cloud reaches the breakdown value $E_{br}$, electrons in air trigger an avalanche that forms a conducting channel. Lightning discharges the accumulated charge, after which the cycle repeats. Rain can carry some charge from the cloud to the ground. This leakage of charge slows down the charge buildup in the cloud, making the lightning less frequent or even stopping it altogether.

In this question we will study the effect of rainfall on the rate of lightning.

Instructions: In parts (a)–(c) only symbolic answers are required; in (d) numerical answers are required.

(a) Rate of charge build up and breakdown. Let $Q$ be the total cloud charge over the footprint $A$. Show that when the breakdown field is reached, the value of the charge is

$Q_{br}={\epsilon }_{0}A{E}_{br}$

Before rainfall started, a lightning was seen roughly every $\Delta t_1=3\text{min}$. Using $Q_{br}$, calculate $I_{int}$, the average rate at which charge builds up internally in the cloud.

(b) Charge carried by rain. Treat a raindrop as a conducting sphere of radius $r_d$ carrying charge $q$, with surface tension $\gamma$ and permittivity ${\epsilon }_0$. The charge a liquid droplet can carry is limited by its surface tension. As charge builds up, electrostatic repulsion pushes the surface outward while surface tension acts to hold it together. Using the interplay of these two forces, show that the largest stable charge — called the Rayleigh charge — is

$q_R=8\pi \sqrt{{\epsilon }_0\gamma r_d^3}$

Hint: You can calculate the electrostatic potential energy of a conducting shell by calculating the work done to add an infinitesimal charge $dq$ to a shell with charge $q$ and integrating. The surface energy is given by $\gamma \times \text{surface area}$. If the total energy of the droplet decreases under an infinitesimal increase in its radius, it is energetically favorable for the droplet to expand or split. Alternatively, consider the balance between the outward electrostatic force on a small surface patch and the inward pull of surface tension along its boundary.

(c) Rain current to ground. Rate of rain $R$ is measured in the height of water that accumulates in a container per hour. If rain falls at rate $R$ (depth per unit time) over area $A$, the volume of water in the rain per unit time is $RA$. The charge appearing on a droplet takes a random value between zero and $q_R$. Assume on average each drop carries only a fraction $\eta$ of the Rayleigh charge, $q_d=\eta q_R$. Show that the downward current to ground is

$I_{rain}=6RA\eta \sqrt{\frac{{\epsilon }_0\gamma }{r_d^3}}$

(d) Lightning during the rainfall. While charge builds up in the cloud at rate $I_{int}$, charge is carried out through the rain at the rate $I_{rain}$. Based on $I_{rain}$ calculated in the previous part, calculate symbolically the time interval $\Delta t_2$ that elapses before the next lightning strikes, once the rainfall begins.

Now use the data below to calculate $\Delta t_2$ numerically. Based on this number, is it likely that one might see a lightning strike after rainfall begins?

Given constants (use orders of magnitude as needed): ${\epsilon }_0=8.85\times 10^{-12}\text{F/m}$, surface tension of water $\gamma =0.072\text{N/m}$, typical raindrop radius $r_d\approx 0.5\text{mm}$, cloud footprint area $A\approx 25{\text{km}}^2$, rainfall rate $R=5\text{mm/hr}$. The air breakdown field is $E_{br}\approx 2.5\times 10^6\text{V/m}$. The fraction $\eta =1/5$.

Topic: Electrostatics, Fluid Mechanics Metodi: Gauss’s Law, Calculus-Integration, Physical Modeling, Differential Equations Competenze: Mathematical Modeling, Physical Reasoning, Estimation & Approximation Fonte: Testo (PDF) — p.20