All practice questions

Question 4341Question →

An electric train starts from rest and accelerates uniformly at $2\text{ m/s}^2$ for a time interval $t$ . It then maintains the maximum velocity attained for a time interval of $t$ . Finally, it decelerates uniformly at $4\text{ m/s}^2$ until it comes to a complete stop. If the total distance covered during the entire journey is $350\text{ m}$ , what is the value of $t$ ?

5\text{ s}

7\text{ s}

10\text{ s}

14\text{ s}

Show answer & explanation

Answer: $10\text{ s}$

Answer

10\text{ s}

The motion consists of three stages: acceleration from rest (

s_1 = t^2

), uniform velocity (

s_2 = 2t^2

), and deceleration to rest (

s_3 = 0.5t^2

). Adding these gives a total distance of

S = 3.5t^2

. Setting

3.5t^2 = 350\text{ m}

yields

t^2 = 100

, so

t = 10\text{ s}

.

Step-by-Step Solution

1

Analyze Stage 1 (Acceleration phase)

Maximum velocity

v = 2t

, displacement

s_1 = \frac{1}{2}(2)t^2 = t^2

Starting from rest (

u = 0

) with acceleration

a_1 = 2\text{ m/s}^2

for duration

t

,

v = u + a_1 t = 2t

and

s_1 = \frac{1}{2} a_1 t^2 = t^2

.

2

Analyze Stage 2 (Constant velocity phase)

Displacement

s_2 = (2t)(t) = 2t^2

The train moves at constant velocity

v = 2t

for duration

t

, so distance is velocity multiplied by time.

3

Analyze Stage 3 (Deceleration phase)

Deceleration time

t_3 = 0.5t

, displacement

s_3 = 0.5t^2

Decelerating from

v = 2t

to

0

at

a_2 = 4\text{ m/s}^2

takes time

t_3 = \frac{v}{a_2} = \frac{2t}{4} = 0.5t

. Distance

s_3 = \frac{1}{2} v t_3 = \frac{1}{2} (2t)(0.5t) = 0.5t^2

.

4

Calculate total displacement and solve for

t

Total distance

S = 3.5t^2 = 350 \implies t^2 = 100 \implies t = 10\text{ s}

Summing the displacements:

S = t^2 + 2t^2 + 0.5t^2 = 3.5t^2

. Equating to

350\text{ m}

gives

3.5t^2 = 350

, so

t^2 = 100

, giving

t = 10\text{ s}

.

Key Concept

Multi-stage linear motion and displacement calculation

Estimated Time:2m 0s

Question 4342Question →

A binary operation $\ast$ is defined on the set of real numbers $\mathbb{R}$ by $a \ast b = a + b - k$ , where $k$ is a constant. If the identity element of the operation is $4$ , what is the inverse of $7$ under this operation?

1

-1

-3

7

Show answer & explanation

Answer: 1

Answer

The inverse of 7 under the binary operation is 1.

First, use the identity property a * e = a with e = 4 to determine that k = 4. Then, set 7 * x = 4 using the operational definition 7 + x - 4 = 4. Solving the linear equation x + 3 = 4 gives the correct inverse value of 1.

Step-by-Step Solution

1

Determine the value of the constant k using the identity element property.

k = 4

By definition of identity element e, a * e = a. Given e = 4, substituting into the definition yields a + 4 - k = a, which simplifies to k = 4.

2

Write the full operational formula.

a * b = a + b - 4

Substitute k = 4 into the original rule a * b = a + b - k.

3

Solve for the inverse element of 7.

x = 1

Let x be the inverse of 7. By definition of inverse, 7 * x = e, so 7 + x - 4 = 4. Simplifying gives x + 3 = 4, hence x = 1.

Key Concept

Identity and inverse elements of a binary operation

Question 4343Question →

In an agricultural survey of $100$ local farmers in a community, $55$ grow Cassava ( $C$ ), $45$ grow Yam ( $Y$ ), and $40$ grow Maize ( $M$ ). It is observed that $20$ farmers grow both Cassava and Yam, $15$ grow both Yam and Maize, and $16$ grow both Cassava and Maize. If $7$ farmers grow all three crops, how many farmers grow none of these three crops?

4

11

16

24

Show answer & explanation

Answer: $4$

Answer

The number of farmers who grow none of the three crops is 4.

The number of farmers who grow at least one crop is found using

n(C \cup Y \cup M) = 55 + 45 + 40 - 20 - 15 - 16 + 7 = 96

. Subtracting this union from the total number of farmers (

100

) gives

100 - 96 = 4

farmers who grow none of the three crops.

Step-by-Step Solution

1

Apply the Principle of Inclusion-Exclusion for three sets to find the total number of farmers growing at least one crop

n(C \cup Y \cup M)

.

n(C \cup Y \cup M) = n(C) + n(Y) + n(M) - n(C \cap Y) - n(Y \cap M) - n(C \cap M) + n(C \cap Y \cap M)

Elements in pairwise intersections are double-counted and elements in all three sets are triple-counted, requiring correction.

2

Substitute the given cardinalities into the formula.

n(C \cup Y \cup M) = 55 + 45 + 40 - 20 - 15 - 16 + 7 = 96

Calculating the total number of elements inside the union of the three sets.

3

Subtract the number of farmers in the union from the universal set size

n(U) = 100

.

n((C \cup Y \cup M)') = 100 - 96 = 4

The complement of the union gives the number of elements belonging to none of the sets.

Key Concept

Principle of Inclusion-Exclusion for Three Sets and Universal Complement

Question 4344Question →

A concave mirror has a radius of curvature of $40\text{ cm}$ . What is the focal length of the mirror?

20\text{ cm}

-20\text{ cm}

40\text{ cm}

80\text{ cm}

Show answer & explanation

Answer: $20\text{ cm}$

Answer

The focal length of the concave mirror is

20\text{ cm}

.

For any spherical mirror, the focal length is half of its radius of curvature (

f = \frac{r}{2}

). Since the mirror is concave, its focal length is positive, yielding

\frac{40\text{ cm}}{2} = 20\text{ cm}

.

Step-by-Step Solution

1

Identify the given physical quantity

Radius of curvature

r = 40\text{ cm}

for a concave mirror.

The radius of curvature is the distance from the pole to the center of curvature.

2

Apply the relationship between focal length and radius of curvature

f = \frac{r}{2} = \frac{40\text{ cm}}{2} = 20\text{ cm}

.

For spherical mirrors, the principal focus lies halfway between the pole and the center of curvature.

Key Concept

Focal length of spherical mirrors

Question 4345Question →

A spherical planet has a radius of $7.2 \times 10^{6} \text{ m}$ and an acceleration due to gravity of $10 \text{ m/s}^2$ at its surface. What is the escape velocity for an object launched from the surface of this planet, in $\text{km/s}$ ?

Show answer & explanation

Answer: 12

Answer

The escape velocity from the surface of the planet is 12 km/s.

The escape velocity

v_e

from the surface of a spherical planet of radius

R

with surface gravity

g

is given by

v_e = \sqrt{2gR}

. Substituting

g = 10 \text{ m/s}^2

and

R = 7.2 \times 10^6 \text{ m}

yields

v_e = \sqrt{2 \times 10 \times 7.2 \times 10^6} = \sqrt{1.44 \times 10^8} = 12000 \text{ m/s}

, which equals

12 \text{ km/s}

.

Step-by-Step Solution

1

Identify the relationship between surface gravity, planetary radius, and escape velocity

The escape velocity formula is

v_e = \sqrt{2gR}

.

Escape velocity is the minimum initial speed required for an object to overcome the gravitational pull of a celestial body.

2

Substitute the given numerical values into the formula

v_e = \sqrt{2 \times 10 \text{ m/s}^2 \times 7.2 \times 10^6 \text{ m}} = \sqrt{144 \times 10^6} \text{ m/s}

.

Plugging in the given values allows direct calculation of the velocity in standard SI units.

3

Simplify the square root and convert units to km/s

v_e = 12000 \text{ m/s} = 12 \text{ km/s}

.

Taking the square root of

144 \times 10^6

gives

12000 \text{ m/s}

, which corresponds to

12 \text{ km/s}

.

Key Concept

Escape Velocity from a Planet's Surface

Question 4346Question →

Match each experimental temperature measurement requirement on the left with the most appropriate thermometric instrument on the right based on its thermometric property and operational characteristics.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct pairings match each measurement requirement to its corresponding thermometric instrument based on its fundamental thermometric property: standard reference calibration pairs with the constant-volume gas thermometer; high-precision steady measurement pairs with the platinum resistance thermometer; rapid localized temperature change measurement pairs with the thermocouple; and non-contact high-temperature measurement pairs with the optical pyrometer.

Each instrument is correctly matched according to the specific physical property that changes measurably with temperature (

P

,

R

, e.m.f., and radiation intensity) and its operational suitability.

Step-by-Step Solution

1

Analyze requirement 1: standard reference calibration using gas pressure at constant volume.

Identified thermometric property as pressure

P

at constant volume

V

, which defines the constant-volume gas thermometer.

Gas thermometers closely approximate the absolute thermodynamic scale and serve as calibration standards.

2

Analyze requirement 2: high-precision steady measurement using resistance variation with thermal lag.

Identified thermometric property as electrical resistance

R

, which corresponds to the platinum resistance thermometer.

Platinum wire resistance changes predictably with temperature, providing high accuracy for stable temperatures.

3

Analyze requirement 3: rapid localized temperature measurement via thermal e.m.f.

Identified thermometric property as thermoelectric voltage (e.m.f.), which corresponds to the thermocouple.

The small thermal mass of thermocouple junctions allows low response times for fast transient measurements.

4

Analyze requirement 4: non-contact measurement of glowing bodies using radiation.

Identified physical principle as thermal radiation intensity, corresponding to the optical pyrometer.

Pyrometers detect infrared/visible radiation, avoiding structural melting associated with direct contact at extreme temperatures.

Key Concept

Thermometric Properties and Operational Limits of Thermometers

Estimated Time:2m 0s

Question 4347Question →

An acoustic pulse generator located at a fixed point between two parallel rigid reflective barriers emits a sound wave with a frequency of $680\text{ Hz}$ . The first echo from the closer barrier is detected after $0.8\text{ s}$ , and the first echo from the farther barrier is detected after $1.4\text{ s}$ . Assuming the speed of sound in the medium is $340\text{ m s}^{-1}$ , what is the total distance between the two barriers, and how many complete wavelengths of this sound wave fit within this total distance?

374\text{ m}

and

748

wavelengths

748\text{ m}

and

1496

wavelengths

102\text{ m}

and

204

wavelengths

374\text{ m}

and

187

wavelengths

Show answer & explanation

Answer: $374\text{ m}$ and $748$ wavelengths

Answer

The total distance between the barriers is

374\text{ m}

and

748

complete wavelengths fit within this distance.

The distance to the first barrier is

d_1 = \frac{340 \times 0.8}{2} = 136\text{ m}

, and to the second barrier is

d_2 = \frac{340 \times 1.4}{2} = 238\text{ m}

. Adding both distances gives a total separation of

374\text{ m}

. With a wavelength

\lambda = \frac{v}{f} = \frac{340}{680} = 0.5\text{ m}

, the number of full wavelengths fitting in

374\text{ m}

is

\frac{374}{0.5} = 748

.

Step-by-Step Solution

1

Calculate the one-way distance from the generator to each reflective barrier using the echo relationship

d = \frac{v \cdot t}{2}

.

For barrier 1:

d_1 = \frac{340 \text{ m s}^{-1} \times 0.8 \text{ s}}{2} = 136 \text{ m}

. For barrier 2:

d_2 = \frac{340 \text{ m s}^{-1} \times 1.4 \text{ s}}{2} = 238 \text{ m}

.

An echo represents a two-way journey (to the surface and back), so the time taken to travel the one-way distance is half of the echo reception time.

2

Determine the total distance between the two parallel barriers.

D = d_1 + d_2 = 136 \text{ m} + 238 \text{ m} = 374 \text{ m}

.

Since the generator is situated between the two barriers, the total separation distance equals the sum of the individual distances to each barrier.

3

Calculate the wavelength

\lambda

of the sound wave using the wave equation

v = f \lambda

.

\lambda = \frac{v}{f} = \frac{340 \text{ m s}^{-1}}{680 \text{ Hz}} = 0.5 \text{ m}$.

Wavelength is the ratio of wave speed to frequency.

4

Compute the number of complete wavelengths

N

contained within the total separation distance.

N = \frac{D}{\lambda} = \frac{374 \text{ m}}{0.5 \text{ m}} = 748.

Dividing total distance by single wavelength yields the number of full wave cycles occupying that span.

Key Concept

Echo distance calculations and wave speed-frequency-wavelength relationships

Estimated Time:2m 0s

Question 4348Question →

A flexible balloon contains a sample of gas occupying a volume of $3.0\text{ m}^3$ at a temperature of $27^\circ\text{C}$ under constant pressure. If the gas is heated to $127^\circ\text{C}$ while keeping the pressure constant, what is the new volume of the gas in $\text{m}^3$ ?

Show answer & explanation

Answer: 4

Answer

The new volume of the gas is

4.0\text{ m}^3

.

According to Charles's Law (

\frac{V_1}{T_1} = \frac{V_2}{T_2}

), at constant pressure, volume is directly proportional to absolute temperature in Kelvin. Converting

27^\circ\text{C}

to

300\text{ K}

and

127^\circ\text{C}

to

400\text{ K}

yields a final volume

V_2 = 3.0 \times \frac{400}{300} = 4.0\text{ m}^3

.

Step-by-Step Solution

1

Convert the given initial and final temperatures from Celsius to Kelvin

T_1 = 27 + 273 = 300\text{ K}

and

T_2 = 127 + 273 = 400\text{ K}

Gas law calculations require absolute temperature in Kelvin.

2

Set up Charles's Law equation for constant pressure processes

\frac{V_1}{T_1} = \frac{V_2}{T_2}

At constant pressure, the volume of a given mass of gas is directly proportional to its absolute temperature.

3

Substitute the values into the equation and solve for the final volume

V_2

V_2 = 3.0 \times \frac{400\text{ K}}{300\text{ K}} = 4.0\text{ m}^3

Direct algebraic evaluation yields the final gas volume.

Key Concept

Charles's Law

Question 4349Question →

A straight conductor of length $0.5\text{ m}$ carries a current of $4.0\text{ A}$ in a uniform magnetic field of flux density $0.20\text{ T}$ . If the conductor is placed at an angle of $30^\circ$ to the magnetic field lines, what is the magnitude of the magnetic force exerted on the conductor?

0.20\text{ N}

0.35\text{ N}

0.40\text{ N}

0.80\text{ N}

Show answer & explanation

Answer: $0.20\text{ N}$

Answer

0.20\text{ N}

The magnetic force on a straight wire carrying current in a uniform magnetic field is given by

F = BIL \sin\theta

. Substituting the given values

B = 0.20\text{ T}

,

I = 4.0\text{ A}

,

L = 0.5\text{ m}

, and

\theta = 30^\circ

yields

F = 0.20 \times 4.0 \times 0.5 \times 0.5 = 0.20\text{ N}

.

Step-by-Step Solution

1

Identify the given physical quantities

L = 0.5\text{ m}

,

I = 4.0\text{ A}

,

B = 0.20\text{ T}

, and

\theta = 30^\circ

.

Extracting values from the problem statement ensures correct substitution.

2

State the magnetic force equation for a straight current-carrying conductor in a magnetic field

F = BIL \sin\theta

Only the component of the magnetic field perpendicular to the current contributes to the magnetic force.

3

Substitute the values and calculate the force magnitude

F = (0.20\text{ T}) \times (4.0\text{ A}) \times (0.5\text{ m}) \times \sin(30^\circ) = 0.20 \times 4.0 \times 0.5 \times 0.5 = 0.20\text{ N}

.

Evaluating

\sin 30^\circ = 0.5

gives the required force value.

Key Concept

Magnetic Force on a Current-Carrying Conductor

Estimated Time:1m 15s

Question 4350Question →

$10$ distinct points are marked on the circumference of a circle. How many different triangles can be formed by connecting any $3$ of these points as vertices?

45

120

360

720

Show answer & explanation

Answer: $120$

Answer

The number of distinct triangles that can be formed is

120

.

To form a triangle, any

3

points must be chosen from the

10

available points. Because no three points on a circle are collinear, every choice of

3

points forms a unique triangle. Since the order of choosing vertices does not alter the triangle, we use combinations:

\binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120

.

Step-by-Step Solution

1

Identify the total number of items

n

and the subset size

r

.

n = 10

points on the circle, and

r = 3

points required to form a triangle.

Any set of

3

non-collinear points uniquely determines a triangle. Since all points lie on a circle, no three points are collinear.

2

Determine whether order matters.

Order does not matter because choosing points

A, B, C

produces the same triangle as choosing

B, C, A

.

Selection of vertices for a geometric shape is a combination problem, not a permutation problem.

3

Apply the combination formula

\binom{n}{r} = \frac{n!}{r!(n - r)!}

.

\binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = \frac{720}{6} = 120

Evaluating the combination gives the total number of distinct triangles.

Key Concept

Combinations for Geometric Formations

Estimated Time:1m 30s

Question 4351Question →

Two long parallel straight conductors separated by a distance of $0.20\text{ m}$ in vacuum carry steady currents of $5.0\text{ A}$ and $8.0\text{ A}$ in opposite directions. Given that the permeability of free space $\mu_0 = 4\pi \times 10^{-7}\text{ T}\cdot\text{m/A}$ , what is the magnitude of the magnetic force per unit length acting between the conductors and the nature of this force?

4.0 \times 10^{-5}\text{ N/m}

, repulsive

4.0 \times 10^{-5}\text{ N/m}

, attractive

8.0 \times 10^{-5}\text{ N/m}

, repulsive

2.0 \times 10^{-5}\text{ N/m}

, attractive

Show answer & explanation

Answer: $4.0 \times 10^{-5}\text{ N/m}$ , repulsive

Answer

4.0 \times 10^{-5}\text{ N/m}

, repulsive

Applying the formula for force per unit length gives

\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d} = \frac{(4\pi \times 10^{-7})(5.0)(8.0)}{2\pi (0.20)} = 4.0 \times 10^{-5}\text{ N/m}

. Since the currents flow in opposite directions, the magnetic interaction is repulsive.

Step-by-Step Solution

1

Identify given variables and the formula for force per unit length between parallel conductors.

Given:

I_1 = 5.0\text{ A}

,

I_2 = 8.0\text{ A}

,

d = 0.20\text{ m}

,

\mu_0 = 4\pi \times 10^{-7}\text{ T}\cdot\text{m/A}

. Formula:

\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}

.

The magnetic force per unit length between two parallel wires depends on the current magnitudes and their distance of separation.

2

Substitute values into the equation and compute the magnitude.

\frac{F}{L} = \frac{(4\pi \times 10^{-7})(5.0)(8.0)}{2\pi (0.20)} = \frac{2 \times 10^{-7} \times 40}{0.20} = 4.0 \times 10^{-5}\text{ N/m}

.

Canceling

4\pi

with

2\pi

yields a multiplier of

2 \times 10^{-7}

, simplifying numerical computation.

3

Determine the direction of force based on current flow direction.

The force is repulsive.

Parallel conductors carrying currents in opposite directions generate magnetic field configurations that force the wires apart.

Key Concept

Magnetic force per unit length between parallel current-carrying conductors

Question 4352Question →

A toy truck of mass $2.5\text{ kg}$ moving at a velocity of $6.0\text{ m s}^{-1}$ collides head-on with a toy car of mass $1.5\text{ kg}$ moving in the opposite direction at $2.0\text{ m s}^{-1}$ . If the two vehicles stick together upon impact, what is their combined velocity immediately after the collision?

3.0\text{ m s}^{-1}

4.5\text{ m s}^{-1}

4.8\text{ m s}^{-1}

2.0\text{ m s}^{-1}

Show answer & explanation

Answer: $3.0\text{ m s}^{-1}$

Answer

3.0\text{ m s}^{-1}

in the direction of the initial motion of the truck

According to the principle of conservation of linear momentum, total initial momentum equals total final momentum. Taking the direction of the toy truck as positive gives an initial momentum of

(2.5\text{ kg} \times 6.0\text{ m s}^{-1}) + (1.5\text{ kg} \times -2.0\text{ m s}^{-1}) = 15.0 - 3.0 = 12.0\text{ kg m s}^{-1}

. Dividing this net momentum by the combined mass

(2.5\text{ kg} + 1.5\text{ kg} = 4.0\text{ kg})

yields a common final velocity of

3.0\text{ m s}^{-1}

.

Step-by-Step Solution

1

Assign direction signs to the velocity vectors

Let the initial direction of the truck be positive (

u_1 = +6.0\text{ m s}^{-1}

). The car moves in the opposite direction, so its velocity is negative (

u_2 = -2.0\text{ m s}^{-1}

).

Linear momentum is a vector quantity, so direction must be accounted for in one-dimensional motion.

2

Calculate the total initial momentum of the system

p_{\text{initial}} = m_1 u_1 + m_2 u_2 = (2.5 \times 6.0) + (1.5 \times -2.0) = 15.0 - 3.0 = 12.0\text{ kg m s}^{-1}

The total momentum before collision is the vector sum of individual momenta.

3

Apply the law of conservation of linear momentum to solve for common final velocity

v

p_{\text{final}} = (m_1 + m_2) v = (2.5 + 1.5) v = 4.0 v

. Setting

p_{\text{final}} = p_{\text{initial}} \Rightarrow 4.0 v = 12.0 \Rightarrow v = 3.0\text{ m s}^{-1}

.

Since the vehicles stick together, they move as a single combined mass.

Key Concept

Conservation of Linear Momentum in Inelastic Collisions

Estimated Time:50s

Question 4353Question →

A meteorological balloon filled with $0.50\text{ kg}$ of helium gas is launched at sea level, where the atmospheric pressure is $1.01 \times 10^5\text{ Pa}$ and the ambient temperature is $27^\circ\text{C}$ . The balloon ascends to a high altitude where the external ambient pressure decreases to $4.00 \times 10^4\text{ Pa}$ and the ambient temperature drops to $-23^\circ\text{C}$ . As the balloon expands, its elastic membrane exerts an additional pressure, causing the internal gas pressure to be $20\%$ higher than the surrounding ambient pressure. Assuming helium behaves as an ideal gas with a molar mass of $4.0\text{ g/mol}$ and the molar gas constant $R = 8.31\text{ J mol}^{-1}\text{K}^{-1}$ , calculate the final volume of helium gas inside the balloon at this altitude in $\text{m}^3$ .

Show answer & explanation

Answer: 5.41

Answer

The final volume of helium gas inside the balloon at altitude is

5.41\text{ m}^3

.

The ideal gas equation

PV = nRT

relates state variables. By determining

n = 125\text{ moles}

from mass and molar mass, absolute temperature

T_2 = 250\text{ K}

, and total internal pressure

P_2 = 4.80 \times 10^4\text{ Pa}

, the final volume

V_2

evaluates to

5.41\text{ m}^3

.

Step-by-Step Solution

1

Convert the mass of helium into moles

n = 125\text{ moles}

Mass

m = 0.50\text{ kg} = 500\text{ g}

divided by molar mass

M = 4.0\text{ g/mol}

gives

n = \frac{500}{4.0} = 125\text{ moles}

.

2

Convert final temperature to Kelvin

T_2 = 250\text{ K}

Gas laws strictly require thermodynamic temperature:

T_2 = -23 + 273 = 250\text{ K}

.

3

Calculate final internal pressure of the gas

P_2 = 4.80 \times 10^4\text{ Pa}

The gas pressure inside the balloon is

20\%

higher than external ambient pressure:

P_2 = 1.20 \times (4.00 \times 10^4) = 4.80 \times 10^4\text{ Pa}

.

4

Solve for the final volume using the ideal gas equation

V_2 = 5.41\text{ m}^3

Rearranging

P_2 V_2 = n R T_2

yields

V_2 = \frac{n R T_2}{P_2} = \frac{125 \times 8.31 \times 250}{4.80 \times 10^4} = 5.41015...\text{ m}^3 \approx 5.41\text{ m}^3

.

Key Concept

Ideal Gas Equation (

PV = nRT

)

Estimated Time:3m 0s

Question 4354Question →

A water tank has a real depth of $16.0\text{ cm}$ . If the refractive index of water relative to air is $\frac{4}{3}$ , what is the apparent depth of the tank in centimetres when viewed normally from above?

Show answer & explanation

Answer: 12

Answer

The apparent depth of the water tank is

12.0\text{ cm}

.

When light passes from water to air, refraction causes the apparent depth to be smaller than the real depth by a factor equal to the refractive index

n

. Substituting a real depth of

16.0\text{ cm}

and

n = \frac{4}{3}

into

\text{Apparent Depth} = \frac{\text{Real Depth}}{n}

yields

12.0\text{ cm}

.

Step-by-Step Solution

1

Identify the relationship between refractive index, real depth, and apparent depth.

n = \frac{\text{Real Depth}}{\text{Apparent Depth}}

Refraction at a plane boundary causes an object immersed in a denser medium to appear closer to the surface when viewed from a rarer medium.

2

Rearrange the equation to isolate Apparent Depth.

\text{Apparent Depth} = \frac{\text{Real Depth}}{n}

The unknown quantity requested by the question is the apparent depth.

3

Substitute the known numerical values into the equation.

\text{Apparent Depth} = \frac{16.0}{\frac{4}{3}} = 16.0 \times \frac{3}{4} = 12.0\text{ cm}

Dividing

16.0

by

\frac{4}{3}

gives

12.0

.

Key Concept

Real and Apparent Depth in Light Refraction

Question 4355Question →

A worker strikes one end of a long solid aluminum pipeline. A detector at the opposite end records two sound signals—one traveling through the aluminum pipeline and the other through the surrounding air—separated by a time interval of $2.8\text{ s}$ . If the speed of sound in air is $340\text{ m s}^{-1}$ and the speed of sound in aluminum is $5100\text{ m s}^{-1}$ , what is the length of the pipeline in meters?

Show answer & explanation

Answer: 1020

Answer

The length of the pipeline is

1020\text{ m}

.

Sound travels significantly faster through solids like aluminum (

5100\text{ m s}^{-1}

) than through gases like air (

340\text{ m s}^{-1}

). The time taken for sound to travel a distance

L

through air is

t_{\text{air}} = \frac{L}{340}

, while through aluminum it is

t_{\text{metal}} = \frac{L}{5100}

. Setting their difference equal to

2.8\text{ s}

gives

\frac{L}{340} - \frac{L}{5100} = 2.8

, which solves to

L = 1020\text{ m}

.

Step-by-Step Solution

1

Formulate transit time expressions for both media

t_{\text{air}} = \frac{L}{340}

and

t_{\text{metal}} = \frac{L}{5100}

Time taken by a wave to travel distance

L

at constant speed

v

is

t = \frac{L}{v}

.

2

Set up the time difference equation

t_{\text{air}} - t_{\text{metal}} = 2.8\text{ s}

The sound wave travels faster through aluminum than air, so the air pulse arrives later by

2.8\text{ s}

.

3

Solve the algebraic equation for distance

L

L = 1020\text{ m}

Combining terms yields

\frac{14L}{5100} = 2.8

, which simplifies to

L = \frac{2.8 \times 5100}{14} = 1020\text{ m}

.

Key Concept

Propagation speed of sound waves in different physical media

Question 4356Question →

According to the kinetic theory of matter, what causes the pressure exerted by a gas on the walls of its container?

The continuous elastic collisions of gas molecules with the walls of the container

The attractive forces pulling gas molecules towards the container walls

The heat capacity of individual gas molecules transferring thermal mass to the walls

The static expansion of fixed gas molecules taking up space

Show answer & explanation

Answer: The continuous elastic collisions of gas molecules with the walls of the container

Answer

The pressure exerted by a gas is due to the continuous elastic collisions of gas molecules with the container walls.

According to the kinetic theory of gases, gas particles are in rapid, random motion. When these particles collide elastically with the walls of the container, they undergo a change in momentum. The average rate of momentum change per unit area exerted by countless particle impacts manifests as macroscopic gas pressure.

Step-by-Step Solution

1

Identify the basic postulate of the kinetic theory of gases regarding particle motion.

Gas molecules are in rapid, constant, and random motion.

Kinetic theory models gas behavior based on particle movement.

2

Relate particle motion to force and pressure on the container boundary.

When particles hit the container wall, they undergo a change in momentum, imparting a force on the wall.

Force is defined as the rate of change of momentum (

F = \frac{\Delta p}{\Delta t}

).

3

Define pressure in terms of force per unit surface area.

The total force exerted per unit surface area by continuous collisions results in gas pressure (

P = \frac{F}{A}

).

Pressure is force distributed over a given surface area.

Key Concept

Kinetic Theory of Matter and Pressure of Gases

Question 4357Question →

Match each electromagnetic device or component listed on the left with its corresponding function or operating principle on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Moving coil galvanometer matches with detecting/measuring small currents via torque; Electric motor matches with converting electrical energy to mechanical energy; Split-ring commutator matches with reversing current every half-cycle to maintain continuous rotation; Soft iron core matches with concentrating magnetic flux to create a radial magnetic field.

Each electromagnetic device or component relies on magnetic forces: moving coil galvanometers convert current to proportional coil torque; electric motors convert electrical energy into continuous mechanical rotation; split-ring commutators reverse current every half-turn to keep torque unidirectionally directed; soft iron cores concentrate magnetic flux to establish strong radial fields.

Step-by-Step Solution

1

Identify the primary operational application of the moving coil galvanometer.

It detects small currents using the torque

T = NIAB \sin\theta

exerted on a current-carrying coil in a magnetic field.

This establishes the link between galvanic deflection and electrical current measurement.

2

Identify the energy transformation principle of an electric motor.

The motor converts input electrical power into mechanical torque via magnetic force

F = BIL\sin\theta

.

This connects the motor to its fundamental mechanical output function.

3

Analyze the mechanical role of a split-ring commutator in DC devices.

It alternates current flow directions in the loop every

180^\circ

.

Without reversals, the coil would oscillate around equilibrium instead of continuously rotating.

4

Determine the ferromagnetic enhancement provided by a soft iron core.

It increases field strength and maintains a radial field orientation.

High magnetic permeability concentrates magnetic field lines effectively.

Key Concept

Operating principles and structural functions of electromagnetic devices based on magnetic torque and forces

Question 4358Question →

An aircraft flies due East at a velocity of $120\text{ m/s}$ while a crosswind blows due North at $50\text{ m/s}$ . What is the magnitude of the resultant velocity of the aircraft?

130\text{ m/s}

170\text{ m/s}

70\text{ m/s}

110\text{ m/s}

Show answer & explanation

Answer: $130\text{ m/s}$

Answer

The magnitude of the resultant velocity of the aircraft is

130\text{ m/s}

.

Because the aircraft's velocity and the wind's velocity are perpendicular to each other, their vector sum is found using the Pythagorean theorem

R = \sqrt{v_1^2 + v_2^2}

. Substituting

120\text{ m/s}

and

50\text{ m/s}

gives

R = \sqrt{120^2 + 50^2} = \sqrt{16900} = 130\text{ m/s}

.

Step-by-Step Solution

1

Identify the given velocity vectors and their relative direction.

Velocity due East

\vec{v}_e = 120\text{ m/s}

and velocity due North

\vec{v}_n = 50\text{ m/s}

are perpendicular to each other (

90^\circ

angle).

East and North directions are orthogonal to each other.

2

Apply the Pythagorean theorem to calculate the magnitude of the resultant vector

R

.

R = \sqrt{v_e^2 + v_n^2} = \sqrt{120^2 + 50^2} = \sqrt{14400 + 2500} = \sqrt{16900} = 130\text{ m/s}

.

The resultant of two perpendicular vectors forms the hypotenuse of a right-angled triangle.

Key Concept

Vector Addition of Perpendicular Vectors

Estimated Time:1m 0s

Question 4359Question →

If $(x, y)$ satisfies the simultaneous equations $x - y = 1$ and $x^2 + y^2 = 25$ , what is the value of the product $xy$ ?

Show answer & explanation

Answer: 12

Answer

12

Expanding

(x - y)^2

yields

x^2 - 2xy + y^2

. Substituting the given values

x - y = 1

and

x^2 + y^2 = 25

into this identity gives

1 = 25 - 2xy

. Rearranging yields

2xy = 24

, which solves to

xy = 12

. Alternatively, solving by substitution gives solution pairs

(4, 3)

and

(-3, -4)

, both yielding a product of

12

.

Step-by-Step Solution

1

Apply the algebraic expansion identity

(x - y)^2 = x^2 + y^2 - 2xy

Connects the difference of terms, the sum of their squares, and their product.

2

Substitute the values given in the system of equations

1^2 = 25 - 2xy

Replaces

x - y

with 1 and

x^2 + y^2

with 25.

3

Isolate and calculate the product

xy

2xy = 24 \implies xy = 12

Simplifies

1 = 25 - 2xy

to find the exact numerical value of

xy

.

Key Concept

Simultaneous Linear and Quadratic Equations

Question 4360Question →

A projectile is launched from ground level over flat terrain with a constant horizontal velocity component of $10\text{ m/s}$ . At a height of $40\text{ m}$ above the ground, the magnitude of its vertical velocity component is equal to its horizontal velocity component. Taking $g = 10\text{ m/s}^2$ , calculate the maximum height reached by the projectile in meters.

Show answer & explanation

Answer: 45

Answer

The maximum height reached by the projectile is

45\text{ m}

.

At the height of

40\text{ m}

, the vertical velocity is equal to the horizontal velocity of

10\text{ m/s}

. Applying the vertical kinematic relation

v_y^2 = u_y^2 - 2gh

yields

10^2 = u_y^2 - 2(10)(40)

, which gives

u_y^2 = 900\text{ m}^2/\text{s}^2

. The maximum height attained above ground level occurs when

v_y = 0

, calculated as

H_{\text{max}} = \frac{u_y^2}{2g} = \frac{900}{20} = 45\text{ m}

.

Step-by-Step Solution

1

Identify the vertical component of velocity at the given height

v_y = 10\text{ m/s}

at

h = 40\text{ m}

The problem states that at

h = 40\text{ m}

, the vertical velocity component equals the constant horizontal component

u_x = 10\text{ m/s}

.

2

Determine the initial vertical launch velocity component

u_y

u_y^2 = 900\text{ m}^2/\text{s}^2 \implies u_y = 30\text{ m/s}

Applying the vertical motion equation

v_y^2 = u_y^2 - 2gh

gives

10^2 = u_y^2 - 2(10)(40) \implies u_y^2 = 100 + 800 = 900

.

3

Calculate the maximum height

H_{\text{max}}

H_{\text{max}} = 45\text{ m}

At maximum height, the vertical velocity becomes zero. Using

H_{\text{max}} = \frac{u_y^2}{2g} = \frac{900}{20} = 45\text{ m}

.

Key Concept

Vertical Kinematics and Maximum Height of a Projectile

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