WAEC SSCE - Physics - 2017

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The main reason why a.c. (alternating current) is transmitted at very high voltage and low current is to reduce the amount of energy lost during transmission. When electricity is transmitted over long distances, some of the energy is lost as heat due to the resistance of the wires. This is known as power loss, and it can be reduced by increasing the voltage of the electricity. By increasing the voltage and decreasing the current, the amount of power loss is reduced because less energy is lost as heat due to the resistance of the wires. Additionally, high voltage a.c. can be stepped down to lower voltages, making it safer for use in homes and businesses. Therefore, transmitting a.c. at very high voltage and low current is more efficient, cost-effective, and reduces heat loss during transmission.

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To solve the problem, we can use the following equation of motion: v^2 = u^2 + 2as where v is the final velocity, u is the initial velocity (which is zero since the body starts from rest), a is the acceleration, and s is the distance travelled. Substituting the given values into the equation, we have: v^2 = 0 + 2(2)(9) v^2 = 36 v = sqrt(36) v = 6.0 ms^-1 Therefore, the final velocity of the body when it has travelled a distance of 9m is 6.0\(ms^{-1}\).

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The maximum density of water occurs at 4°C (39°F). This is because water is most dense at this temperature and any change in temperature either way will cause it to become less dense. Imagine a glass of ice water and a glass of hot water. The ice water is less dense than the water it was made from because the ice takes up more space than the water it came from. On the other hand, hot water is less dense than the ice water because its molecules are moving around more, causing them to take up more space. But at 4°C, water reaches its maximum density because the heat causes the water molecules to move around just enough to make it dense, but not so much that they take up more space.

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The inverse of the time required for a wave to complete one full cycle is called the "frequency". Frequency refers to how many cycles of a wave occur in a given amount of time. It is usually measured in Hertz (Hz), which is the number of cycles per second. So, the higher the frequency of a wave, the more cycles occur in a given amount of time, and vice versa. To understand this concept, imagine you are watching waves in the ocean. The period of the wave would be the time it takes for one complete wave to pass by you, while the frequency would be the number of waves that pass by you in a given amount of time (e.g. one minute). So, if the waves are coming in quickly, with a high frequency, you will see many waves passing by in a short amount of time. On the other hand, if the waves are coming in slowly, with a low frequency, you will see fewer waves passing by in the same amount of time. Overall, frequency is an important property of waves that helps us understand their behavior and characteristics.

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The force per unit charge experienced at a point in a field is the electric field intensity. This means that if you were to place a tiny positive test charge at that point in the field, the electric field intensity would tell you how much force that charge would experience. Essentially, the electric field intensity describes how strong the electric field is at that particular point. It is important to note that this only applies to electric fields, not gravitational or magnetic fields.

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Gamma radiation cannot be seen by the naked eye and can pass through many materials, including an aluminum foil. Gamma radiation is a type of high-energy electromagnetic radiation and can be present naturally in our surroundings, as well as being produced artificially in some industrial and medical applications. Gamma radiation can also be used as a tracer, for example, in medical imaging to visualize internal organs and tissues.

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To calculate the cost of lighting the lamp, we need to determine the amount of energy consumed by the lamp, which is given by: Energy = Power x Time Where power is given in Watts, time is given in hours, and energy is given in kilowatt-hours (kWh). First, we need to convert the power rating of the lamp from 100W to kW: Power = 100W / 1000 = 0.1 kW Next, we can calculate the energy consumed by the lamp during the 5 hours it was lit: Energy = Power x Time = 0.1 kW x 5 hours = 0.5 kWh Finally, we can calculate the cost of lighting the lamp based on the cost of 1 kWh of electrical energy, which is given as N5: Cost = Energy x Cost per kWh = 0.5 kWh x N5/kWh = N2.50 Therefore, the cost of lighting the lamp is N2.50. Option (A) N2.50 is the correct answer.

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The output voltage of a transformer can be calculated using the formula: output voltage = input voltage × (number of turns in secondary coil / number of turns in primary coil) In this case, the input voltage is given as 240\(V_{r.m.s}\), the number of turns in the primary coil is 3000, and the number of turns in the secondary coil is 500. So, the output voltage can be calculated as: output voltage = 240\(V_{r.m.s}\) × (500/3000) = 40\(V_{r.m.s}\) Therefore, the output voltage of the transformer is 40\(V_{r.m.s}\). Answer option C is correct. Note that the root mean square (RMS) value of an AC voltage represents the equivalent DC voltage that would produce the same heating effect in a resistor. In other words, an RMS voltage of 240\(V_{r.m.s}\) would produce the same heating effect as a DC voltage of 240V.

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The energy stored in a spring can be calculated using the formula: Energy (E) = 0.5 * k * x^2 where k is the stiffness constant of the spring and x is the amount it is extended or compressed from its rest position. In this case, k = 2000 Nm^-1 and x = 0.04 m (4 cm converted to meters). Substituting the values into the formula: E = 0.5 * 2000 * 0.04^2 E = 0.5 * 2000 * 0.0016 E = 1.60 J So, the energy stored in the spring when it is extended 4 cm is 1.60 J.

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In the given wave equation \(y = 10sin(x - 50t)\), the number 10 represents the amplitude of the wave. The amplitude is the maximum displacement of a wave from its rest position. In other words, it is the height of the wave from the middle or equilibrium point to the highest point or the lowest point. In this case, the amplitude of the wave is 10, which means that the wave oscillates between the values of +10 and -10 units from its equilibrium position. Therefore, the amplitude of a wave determines its intensity or strength.

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The reason for the observation that the bubbles get bigger as they rise is because the atmospheric pressure on the bubbles decreases. When water is heated, the temperature of the water increases, causing the air and other gases dissolved in the water to become less soluble and form bubbles. As these bubbles rise to the surface of the water, the pressure on them decreases, causing the bubbles to expand and become bigger in size. This is because the atmospheric pressure decreases with height and the pressure inside the bubble decreases as the bubble rises to a lower pressure environment. This causes the air or gas inside the bubble to expand, making the bubble bigger.

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To completely melt the ice, we need to supply heat energy to it equal to its mass multiplied by its specific latent heat of fusion. So, the heat energy required to melt the 50g of ice completely is given by: Heat energy required = mass × specific latent heat of fusion = 50g × 336 J/g = 16800 J But we know that only 70% of the total heat energy supplied was used to completely melt the ice. So, we can set up the following equation: Total heat energy supplied × 70/100 = 16800 J Simplifying the equation, we get: Total heat energy supplied = 16800 J × 100/70 = 24000 J Therefore, the total heat energy supplied is 24000 J. Answer option D is correct.

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The angle at which a projectile must be fired to cover maximum range is 45 degrees. When an object is thrown or fired, it follows a curved path known as a projectile motion. The path of the projectile is affected by gravity and air resistance. To achieve maximum range, the projectile must be launched at an angle that balances the effects of gravity and air resistance. If the projectile is launched at a very low angle (close to 0 degrees), it will not travel very far because gravity will pull it down quickly. On the other hand, if it is launched at a very high angle (close to 90 degrees), it will not travel very far because most of its initial velocity will be directed upwards against gravity. At 45 degrees, the projectile's initial velocity is divided equally into its horizontal and vertical components. The horizontal component of velocity keeps the projectile moving forward while the vertical component of velocity is affected by gravity, causing the projectile to rise and fall. This balance between horizontal and vertical velocities allows the projectile to cover the maximum distance before hitting the ground. Therefore, firing a projectile at 45 degrees will give it the maximum range.

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The distance covered by the motorist can be calculated using the formula: distance = speed x time We are given that the speed of the motorist is 72 km/h, but we need to convert it to meters per second (m/s) to be consistent with the time interval given in seconds. 1 km/h = 0.27778 m/s (approx.) So, speed = 72 km/h x 0.27778 m/s/km/h = 20 m/s (approx.) The time interval given is 0.4 s. Putting these values in the formula: distance = speed x time distance = 20 m/s x 0.4 s = 8 m (approx.) Therefore, the distance covered by the motorist during the 0.4s with eyes shut is approximately 8 meters. Hence, the correct option is (D) 8.0m.

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The Heisenberg uncertainty principle states that it is impossible to precisely determine both the position and momentum of a particle at the same time. Complementary variables are pairs of physical quantities that are related by the uncertainty principle. Out of the options given, the pair that constitutes complementary variables is "position and momentum." This means that the more accurately you measure the position of a particle, the less precisely you can determine its momentum, and vice versa. To give a simple example, imagine trying to determine the position and momentum of a ball. If you know the position of the ball very precisely, you would have to give up some information about its momentum, and if you know its momentum very precisely, you would have to give up some information about its position. This is a fundamental property of the universe that arises from the wave-like nature of particles at the quantum level.

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The object that is not a conductor of electricity is "Dry wood." A conductor of electricity is a material that allows electrical current to flow through it easily. Materials like metals, such as iron, are good conductors because they have a lot of free electrons that can carry electric charge. However, dry wood is not a good conductor of electricity. This is because wood is an insulator and does not have free electrons that can carry electrical charge. In summary, the Earth, human body, and iron rod are all conductors of electricity, but dry wood is not.

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The statement "Molecules of solids move more freely than molecules of liquids and gases" is not correct. In fact, molecules of solids are held tightly in a fixed position, while molecules of liquids and gases have more freedom of movement. In a solid, the molecules are tightly packed together and held in a fixed position by intermolecular forces. This is why solids have a fixed shape and volume. In liquids, the molecules are still held together by intermolecular forces, but they have more freedom to move around and slide past each other. This is why liquids have a fixed volume but not a fixed shape. In gases, the molecules are widely spaced apart and move freely in all directions. This is why gases have neither a fixed shape nor a fixed volume. The other statements are all correct. Molecules of solids are more closely packed than those of liquids, energy is required to break the intermolecular forces of attraction between molecules, and molecules of liquids move about within the liquid and they are in constant motion.

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Out of the given options, both Condensation and Evaporation are surface phenomena, while Photo Emission and Thermionic Emission are not. Condensation is the process of converting a gas or vapor into a liquid by cooling it down. This process occurs on the surface of a solid or liquid and is a surface phenomenon. Evaporation is the process of converting a liquid into a gas or vapor by heating or increasing its surface area. This process also occurs on the surface of a solid or liquid and is also a surface phenomenon. Photo Emission and Thermionic Emission, on the other hand, are not surface phenomena. Photo Emission is the process of electrons being emitted from a material when it absorbs light of a certain frequency. This process occurs in the bulk of the material and not just on the surface. Thermionic Emission is the process of electrons being emitted from a heated material due to its high temperature. This process also occurs in the bulk of the material and not just on the surface. In summary, Condensation and Evaporation are surface phenomena, while Photo Emission and Thermionic Emission are not.

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When two waves of equal amplitude and frequency approach each other, they interfere with each other. The interference can be constructive or destructive, depending on the phase difference between the waves. In this particular scenario, the two waves have the same amplitude and frequency, and they meet at a point I midway between them. At this point, the two waves are in phase with each other, which means that the crest of one wave coincides with the crest of the other wave, and the same is true for the troughs. When two waves are in phase with each other, they reinforce each other, resulting in constructive interference. The amplitude of the resulting wave is the sum of the amplitudes of the two waves. In this case, the resulting wave will have an amplitude of 2A, which is twice the amplitude of each individual wave. Therefore, the answer is 2A, as the resulting wave will have an amplitude of twice the amplitude of each individual wave due to constructive interference.

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The efficiency of a machine is the ratio of useful output energy to the input energy. In this case, the useful output energy is the energy used to grind corn, and the input energy is the electrical energy used to power the motor. The problem states that one-quarter of the energy is wasted on sound and heat. Therefore, the useful output energy is three-quarters (i.e., 100% - 25%) of the input energy. We can calculate the useful output energy as follows: Useful output energy = 3/4 x 1.44 x 10^5 J = 1.08 x 10^5 J Now, we can calculate the efficiency of the machine as follows: Efficiency = (Useful output energy / Input energy) x 100% = (1.08 x 10^5 J / 1.44 x 10^5 J) x 100% = 0.75 x 100% = 75% Therefore, the efficiency of the corn mill machine is 75%.

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The device that converts light energy to electrical energy is a solar cell. Solar cells are made up of materials that absorb photons (particles of light) and release electrons, generating an electrical current. When sunlight hits the solar cell, the photons knock the electrons loose from their atoms, allowing them to flow through the material to produce electricity. This process is known as the photovoltaic effect. Bulbs, televisions, and LEDs do not convert light energy to electrical energy; they all use electrical energy to produce light.

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The term being described in this question is "saturated." When a vapor is in dynamic equilibrium with its own liquid, it means that there is a balance between the number of molecules evaporating from the liquid and the number of molecules condensing back into the liquid. This balance happens when the vapor pressure of the liquid reaches a certain level, known as the "saturation vapor pressure." At this point, the vapor is said to be "saturated." So, to summarize: a vapor in dynamic equilibrium with its own liquid is saturated because the number of molecules evaporating from the liquid is balanced by the number of molecules condensing back into the liquid, resulting in a certain vapor pressure known as the saturation vapor pressure.

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The thermal energy that is added or removed from a substance causing it to change its physical state (such as from solid to liquid or from liquid to gas) is called "latent heat." This term refers to the fact that the heat is "hidden" or "latent" within the substance and does not cause a temperature change, but instead is used to change the molecular arrangement of the substance. For example, when ice is melted into water, the heat added is used to break the bonds between the ice molecules and allow them to move more freely, rather than increasing the temperature of the water. Latent heat is an important concept in thermodynamics and is used to explain many natural phenomena, such as the formation of clouds, the melting of glaciers, and the evaporation of water.

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The nucleon number of an atom is the total number of protons and neutrons in the nucleus of an atom. This number represents the atomic mass of the atom and is unique for each element. The number of protons determines the atomic number of an element and therefore its identity, as every element has a unique atomic number. The number of neutrons in an atom can vary, leading to different isotopes of the same element.

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The phenomenon where plane waves, such as those produced by a small gap, spread out and form circular waves is known as diffraction. Diffraction occurs when a wave passes through an opening or obstacle that is comparable in size to its wavelength. The waves spread out in all directions after passing through the gap, and their shape becomes more circular as they move further away from the gap. Interference is the phenomenon where two or more waves combine to form a resultant wave, which can be constructive or destructive. Dispersion is the process where different wavelengths of light bend at different angles when passing through a medium, resulting in a separation of colors. Refraction is the bending of waves when they pass from one medium to another due to a change in their speed.

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The ionization energy of an atom is the minimum amount of energy required to completely remove an electron from the atom, resulting in the formation of a positively charged ion. In the table above, the energy level of the ground state of mercury is -10.4eV, which means that it would require at least +10.4eV of energy to completely remove an electron from the atom. To calculate the ionization energy, we need to find the difference between the energy level of the highest occupied electron (in this case, n = alpha) and the energy level of the ground state (n=0), which can be expressed as: ionization energy = energy of highest occupied level - energy of ground state Using the values given in the table, we get: ionization energy = 0.06eV - (-10.4eV) = 10.46eV Therefore, the ionization energy of the mercury atom is approximately +10.46eV, which is closest to option B (+10.4eV).

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The potentiometer wire is a wire of known resistance per unit length, and it acts as a potential divider. The battery of e.m.f 3.0V is connected across the wire AB, and the jockey J makes contact at the 30cm mark, which means that the wire AJ is 30cm long and wire BJ is 70cm long. The resistance of the wire AJ can be calculated as follows: Resistance of AJ = Resistance per unit length * Length of AJ = 10\(\Omega m^{-1}\) * 0.3m = 3.0\(\Omega\) The voltage across AJ can be calculated by using the concept of a potential divider. The voltage across AJ can be given by: Voltage across AJ = Total voltage * (Resistance of AJ / Total resistance) = 3.0V * (3.0\(\Omega\) / (3.0\(\Omega\) + 10\(\Omega m^{-1}\) * 0.7m)) = 0.9V So, the resistance of length AJ is 3.0\(\Omega\) and the voltage across AJ is 0.9V.

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The upthrust on an object partially or fully immersed in a fluid is equal to the weight of the fluid displaced by the object. This is known as Archimedes' principle. In this case, the mass of water displaced by the object is given as 200g, which is equivalent to 0.2kg (since 1kg = 1000g). To calculate the weight of the water displaced, we need to multiply the mass by the acceleration due to gravity, g, which is given as 10m/s² in the question. Therefore, weight of water displaced = 0.2kg x 10m/s² = 2N Since the upthrust on the object is equal to the weight of the water displaced, the upthrust on the object is 2N. Hence, the answer is option (A) 2.0N.

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For a particle dropped from rest, the initial velocity, \(u=0\), and the acceleration is \(g\).

Using the equation of motion,

\[h=ut+\frac{1}{2}gt^2\]

\[h=\frac{1}{2}gt^2\]

Thus, \(h\) is directly proportional to \(t^2\). A plot of \(h\) on the vertical axis against \(t^2\) on the horizontal axis is a straight line passing through the origin.

The gradient of the graph is

\[\text{gradient}=\frac{h}{t^2}=\frac{g}{2}.\]

Hence, \(g=2\times\text{gradient}\).

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Three advantages of fluorescent tubes over filament (incandescent) bulbs

1. Higher luminous efficiency: a fluorescent tube produces much more light for the same electrical power consumed, so it is more economical to run.
2. Less heat is produced: most of the energy is converted to light rather than wasted as heat, so the tube runs cool.
3. Longer working life: a fluorescent tube lasts far longer than a filament bulb because it has no fragile hot filament to burn out.

(Other acceptable points: the light is more evenly spread and softer on the eyes, and the running cost is lower.)

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Phenomena explained by the molecular (kinetic) theory of matter

• Diffusion - the spreading of one substance through another, due to the random motion of molecules.
• Brownian motion - the ceaseless, random, zig-zag movement of tiny particles (such as smoke or pollen grains) caused by bombardment by fast-moving air or liquid molecules.
• Evaporation - the escape of the more energetic molecules from a liquid surface, accompanied by cooling of the liquid left behind.

(Surface tension, capillarity, expansion on heating and osmosis are also acceptable.)

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(a) (i) Nuclear fission is the splitting of a heavy atomic nucleus into two lighter nuclei, with the release of a large amount of energy and neutrons.

(ii) Functions of materials in a nuclear fission reactor

Material	Function
Graphite	It acts as a moderator, slowing down fast neutrons so that they can cause further fission.
Boron rods	They absorb excess neutrons and hence control the rate of the chain reaction.
Liquid sodium	It acts as a coolant, removing heat from the reactor core.

(b) (i) Energy-level diagram of the hydrogen atom:

(ii) For the transition from \(n=3\) to the ground state \(n=1\),

\[\Delta E=E_3-E_1=(-1.51)-(-13.60)=12.09\ \text{eV}\]

\[\Delta E=12.09\times1.6\times10^{-19}=1.9344\times10^{-18}\ \text{J}\]

Using \(\Delta E=\dfrac{hc}{\lambda}\),

\[\lambda=\frac{hc}{\Delta E}=\frac{(6.6\times10^{-34})(3.0\times10^8)}{1.9344\times10^{-18}}\]

\[\lambda=1.02\times10^{-7}\ \text{m}\]

The wavelength of the emitted photon is \(\boxed{1.02\times10^{-7}\ \text{m}}\), or \(102\ \text{nm}\).

(c) The activity is proportional to the number of undecayed carbon-14 nuclei. Thus,

\[\frac{A}{A_0}=\frac{9.5}{12.0}=e^{-\lambda t}\]

\[\lambda=\frac{0.693}{5572}=1.244\times10^{-4}\ \text{year}^{-1}\]

\[t=\frac{1}{\lambda}\ln\left(\frac{A_0}{A}\right)=\frac{1}{1.244\times10^{-4}}\ln\left(\frac{12.0}{9.5}\right)\]

\[t=1.88\times10^3\ \text{years}\]

Therefore, the age of the ancient bone is \(\boxed{1.88\times10^3\ \text{years}}\), approximately \(\boxed{1878\ \text{years}}\).

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Three advantages of a p-n junction (semiconductor) diode over a diode valve (vacuum-tube diode)

1. Much smaller and lighter: a semiconductor diode is very compact, so circuits built with it can be made small and portable.
2. No heater/filament needed: it works cold and requires no warming-up time, whereas a diode valve needs a heated cathode; the semiconductor diode therefore consumes far less power.
3. More rugged, cheaper and longer-lasting: it has no fragile glass envelope or filament to break or burn out, so it is mechanically robust, inexpensive and durable.

(Other acceptable points: it operates at lower voltages and switches on instantly.)

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Given: a load of 4 N makes the spring 14 cm long; a load of 6 N produces an extension of 4 cm. Let the unstretched (natural) length be \(L_0\) and the force constant be k.

By Hooke's law, extension is proportional to load, so the force constant is found from the second statement:

\[ k = \frac{F}{e} = \frac{6\ \text{N}}{4\ \text{cm}} = 1.5\ \text{N cm}^{-1} \]

Extension produced by the 4 N load:

\[ e_1 = \frac{F}{k} = \frac{4}{1.5} = 2.67\ \text{cm} \]

The length under the 4 N load is 14 cm, and this equals natural length plus extension:

\[ L_0 = 14 - e_1 = 14 - 2.67 = 11.33\ \text{cm} \]

The unstretched length of the spring is approximately 11.3 cm.

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(a) Two factors on which surface tension depends

• Temperature of the liquid: surface tension decreases as the temperature rises (the molecules gain more energy and their mutual attraction weakens).
• Nature of the liquid and any dissolved substances (impurities): different liquids have different surface tensions, and adding impurities such as detergent or soap lowers it, while some dissolved salts raise it.

(b) Making mosquito larvae sink

Mosquito larvae hang from the underside of the water surface and breathe there, being supported by the surface tension of the water. If a thin film of oil (or kerosene, or a little detergent) is spread over the stagnant water, the surface tension is greatly reduced. The surface can then no longer support the larvae, so they lose their hold on the surface film and sink. The oil film also blocks their breathing tubes, so they are unable to take in air.

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(a)(i) Dielectric

A dielectric is an insulating material (solid, liquid or gas) placed between the two plates of a capacitor. When placed in the field it becomes polarised and increases the capacitance of the capacitor. Examples are mica, waxed paper and air.

(a)(ii) Capacitance

\[ C = \frac{\varepsilon_r\,\varepsilon_0\,A}{d} \]

with \(\varepsilon_r = 900\), \(\varepsilon_0 = 8.85\times10^{-12}\,\text{Fm}^{-1}\), \(A = 9.6\times10^{-2}\,\text{m}^2\) and \(d = 2.25\times10^{-3}\,\text{m}\):

\[ C = \frac{900 \times 8.85\times10^{-12} \times 9.6\times10^{-2}}{2.25\times10^{-3}} = 3.4\times10^{-7}\,\text{F} \]

(b)(i) Ammeter or voltmeter?

The voltmeter has the higher resistance. A voltmeter is connected in parallel with the component whose p.d. is being measured, so it must have a very high resistance to draw only a negligible current and avoid disturbing the circuit. An ammeter is connected in series and must have a very low resistance so that it does not alter the current it measures.

(b)(ii) Voltmeter reading

The two 400 \(\Omega\) resistors on the left are in parallel:

\[ \frac{1}{R_1} = \frac{1}{400} + \frac{1}{400} \Rightarrow R_1 = 200\,\Omega \]

The 800 \(\Omega\) resistor is in parallel with the 800 \(\Omega\) voltmeter:

\[ \frac{1}{R_2} = \frac{1}{800} + \frac{1}{800} \Rightarrow R_2 = 400\,\Omega \]

\(R_1\) and \(R_2\) are in series across the 6 V supply:

\[ R_T = 200 + 400 = 600\,\Omega, \qquad I = \frac{V_T}{R_T} = \frac{6}{600} = 0.01\,\text{A} \]

The voltmeter reads the p.d. across \(R_2\):

\[ V = I\,R_2 = 0.01 \times 400 = 4.0\,\text{V} \]

(c) Equivalent resistance

The two 2 \(\Omega\) resistors in parallel combine to:

\[ \frac{1}{R_p} = \frac{1}{2} + \frac{1}{2} = 1 \Rightarrow R_p = 1\,\Omega \]

This 1 \(\Omega\) is in series with the remaining 2 \(\Omega\) and 2 \(\Omega\) resistors:

\[ R_T = 2 + 1 + 2 = 5\,\Omega \]

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Given: horizontal velocity \(u = 15\ \text{m s}^{-1}\), height \(h = 20\ \text{m}\), \(g = 10\ \text{m s}^{-2}\).

The horizontal and vertical motions are independent. First find the time of flight from the vertical motion (initial vertical velocity is zero):

\[ h = \tfrac{1}{2}gt^2 \Rightarrow 20 = \tfrac{1}{2}\times10\times t^2 = 5t^2 \] \[ t^2 = 4 \Rightarrow t = 2\ \text{s} \]

The horizontal velocity stays constant, so the horizontal distance (range) is:

\[ R = u \times t = 15 \times 2 = 30\ \text{m} \]

The particle covers a horizontal distance of 30 m before hitting the ground.

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(a) A wavefront is a surface or locus of points in a wave which are vibrating in the same phase.

(b)(i) Two practical uses of glass prisms are:

1. to change the direction of a beam of light, for example in binoculars or periscopes;
2. to disperse white light into its component colours, for example in a spectrometer.

(b)(ii) The deviation produced by a triangular glass prism depends on:

1. the angle of incidence of the ray;
2. the refractive index of the glass prism.

The refracting angle of the prism is also a factor.

(b)(iii) The graph of angle of deviation, d, against angle of incidence, i, is shown below. The minimum point is the point at which i = e.

(c)(i) Astronomical telescope in normal adjustment:

In normal adjustment, the real intermediate image formed by the objective lies at the first focal point of the eyepiece. The emergent rays are parallel and the final image is therefore at infinity.

(c)(ii)

For an astronomical telescope in normal adjustment,

\[M=\frac{f_o}{f_e}\]

Given \(M=5\) and \(f_o=100\text{ cm}\):

(α) Focal length of eyepiece

\[5=\frac{100}{f_e}\]

\[f_e=\frac{100}{5}=20\text{ cm}\]

(β) Length of telescope

\[L=f_o+f_e=100+20=120\text{ cm}\]

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(a) Stable equilibrium

A rigid body is in stable equilibrium if, when it is slightly displaced and released, it tends to return to its original position. Its centre of gravity rises on displacement, so that its potential energy is a minimum at the equilibrium position.

(b) Block and tackle of velocity ratio 3

The rope is fixed to the movable block. It then passes over the left fixed pulley, round the movable pulley, and over the right fixed pulley. Thus, the movable block is supported by three tensioned sections of the rope, giving:

\[\text{V.R.}=3\]

(c) Temperature of the gas at the end of the race

Since the volume of the tyre is constant,

\[\frac{P_1}{T_1}=\frac{P_2}{T_2}\]

\[P_1=4.5\times10^5\ \text{Pa},\qquad P_2=4.6\times10^5\ \text{Pa}\]

\[T_1=20^\circ\text{C}+273=293\ \text{K}\]

\[T_2=\frac{P_2T_1}{P_1}=\frac{(4.6\times10^5)(293)}{4.5\times10^5}=299.5\ \text{K}\]

\[\boxed{T_2=299.5\ \text{K}\approx 26.5^\circ\text{C}}\]

(d)(i) Temperature scales

The calibration readings are:

Fixed point	Temperature	Resistance	Gas pressure
Ice point	273 K	5.67 \(\Omega\)	\(7.13\times10^4\) Pa
Steam point	373 K	7.75 \(\Omega\)	\(9.74\times10^4\) Pa

Let \(R_\theta\) be the resistance of the platinum thermometer in the liquid bath and \(P_\theta\) be the pressure of the gas thermometer in the same bath.

(α) Platinum resistance thermometer

\[\frac{R_\theta-5.67}{7.75-5.67}=\frac{T_R-273}{373-273}\]

\[\boxed{T_R=273+\frac{100(R_\theta-5.67)}{2.08}\ \text{K}}\]

(β) Constant-volume gas thermometer

\[\frac{P_\theta-7.13\times10^4}{(9.74-7.13)\times10^4}=\frac{T_G-273}{373-273}\]

\[\boxed{T_G=273+\frac{100(P_\theta-7.13\times10^4)}{2.61\times10^4}\ \text{K}}\]

(d)(ii) Percentage error of the platinum resistance thermometer

Taking the constant-volume gas thermometer temperature as the reference temperature,

\[\boxed{\%\,\text{error}=\frac{T_R-T_G}{T_G}\times100\%}\]

where \(T_R\) and \(T_G\) are obtained from the two boxed calibration equations above.

Maelezo ya Majibu

(a) Deductions from a displacement-time graph

1. The velocity is obtained from the gradient of the graph. For the straight line shown,

\[v=\frac{\Delta s}{\Delta t}=\frac{10-2}{4-0}=2.0\ \text{m s}^{-1}.\]

2. The initial displacement is given by the intercept on the displacement axis. From the graph, the initial displacement is \(2.0\ \text{m}\). A straight line also indicates uniform velocity.

(b) Effect on the gravitational force

The gravitational force between two masses is

\[F=\frac{Gm_1m_2}{r^2}.\]

When each mass and the distance between them are doubled,

\[F' = \frac{G(2m_1)(2m_2)}{(2r)^2}=\frac{4Gm_1m_2}{4r^2}=\frac{Gm_1m_2}{r^2}=F.\]

Therefore, the force between the masses remains unchanged.

(c) Factors affecting the maximum height attained by a fired bullet

1. The initial speed, or muzzle velocity, of the bullet.
2. The angle of projection of the bullet.

Other acceptable factors include acceleration due to gravity and air resistance.

(d) Practical examples of mechanical resonance

1. A glass may shatter when subjected to a sound whose frequency equals the natural frequency of the glass.
2. A bridge may vibrate dangerously when soldiers march across it in step.

Time taken by the body to reach the bottom of the plane

Given: \(h=4.0\ \text{m}\), \(\theta=30^\circ\), \(\mu=0.3\), \(u=0\), and \(g=10\ \text{m s}^{-2}\).

The length \(s\) of the inclined plane is

\[s=\frac{h}{\sin 30^\circ}=\frac{4.0}{0.5}=8.0\ \text{m}.\]

The acceleration down the plane is

\[a=g(\sin\theta-\mu\cos\theta)\]

\[a=10\left(\sin30^\circ-0.3\cos30^\circ\right)\]

\[a=10\left(0.5-0.3\times0.866\right)=2.40\ \text{m s}^{-2}.\]

Using \(s=ut+\tfrac12at^2\),

\[8.0=0+\frac12(2.40)t^2\]

\[t^2=\frac{16.0}{2.40}=6.67\]

\[t=2.58\ \text{s}\approx2.6\ \text{s}.\]

Hence, the body takes \(2.6\ \text{s}\) to reach the bottom of the plane.