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(a) Define velocity.
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(b) The drag force $F_D$ acting on a car moving with speed $v$ along a straight horizontal road is given by
$$F_D = v^2Ak$$
where $k$ is a constant and $A$ is the cross-sectional area of the car.
Determine the SI base units of $k$.
SI base units .......................................................... [2]
(c) The value of $k$, in SI base units, for the car in (b) is 0.24. The cross-sectional area $A$ of the car is 5.1 m$^2$.
The car is travelling with a constant speed along a straight road and the output power of the engine is $4.8 \times 10^4$ W. Assume that the output power of the engine is equal to the rate at which the drag force $F_D$ is doing work against the car.
Determine the speed of the car.
speed = ..................................................... m s$^{-1}$ [3]
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(a) Fig. 2.1 shows the velocity–time graph for an object moving in a straight line.
(i) Determine an expression, in terms of $u$, $v$ and $t$, for the area under the graph. [1]
(ii) State the name of the quantity represented by the area under the graph. [1]
(b) A ball is kicked with a velocity of 15 m s$^{-1}$ at an angle of 60° to horizontal ground. The ball then strikes a vertical wall at the instant when the path of the ball becomes horizontal, as shown in Fig. 2.2.
Assume that air resistance is negligible.
(i) By considering the vertical motion of the ball, calculate the time it takes to reach the wall. [3]
(ii) Explain why the horizontal component of the velocity of the ball remains constant as it moves to the wall. [1]
(iii) Show that the ball strikes the wall with a horizontal velocity of 7.5 m s$^{-1}$. [1]
(c) The mass of the ball in (b) is 0.40 kg. It is in contact with the wall for a time of 0.12 s and rebounds horizontally with a speed of 4.3 m s$^{-1}$.
(i) Use the information from (b)(iii) to calculate the change in momentum of the ball due to the collision. [2]
(ii) Calculate the magnitude of the average force exerted on the ball by the wall. [1]
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(a) Explain what is meant by work done.
(b) A ball of mass 0.42 kg is dropped from the top of a building. The ball falls from rest through a vertical distance of 78 m to the ground. Air resistance is significant so that the ball reaches constant (terminal) velocity before hitting the ground. The ball hits the ground with a speed of 23 m s⁻¹.
(i) Calculate, for the ball falling from the top of the building to the ground:]
1. the decrease in gravitational potential energy [2]
2. the increase in kinetic energy. [2]
(ii) Use your answers in (b)(i) to determine the average resistive force acting on the ball as it falls from the top of the building to the ground. [2]
(c) The ball in (b) is dropped at time $t = 0$ and hits the ground at time $t = T$. The acceleration of free fall is $g$.
On Fig. 3.1, sketch a line to show the variation of the acceleration $a$ of the ball with time $t$ from time $t = 0$ to $t = T$.
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(a) State the difference between progressive waves and stationary waves in terms of the transfer of energy along the wave. [1]
(b) A progressive wave travels from left to right along a stretched string. Fig. 4.1 shows part of the string at one instant.
P, Q and R are three different points on the string. The distance between P and R is 0.48m. The wave has a period of 0.020s.
(i) Use Fig. 4.1 to determine the wavelength of the wave. [1]
(ii) Calculate the speed of the wave. [2]
(iii) Determine the phase difference between points Q and R. [1]
(iv) Fig. 4.1 shows the position of the string at time $t = 0$. Describe how the displacement of point Q on the string varies with time from $t = 0$ to $t = 0.010s$. [2]
(c) A stationary wave is formed on a different string that is stretched between two fixed points X and Y. Fig. 4.2 shows the position of the string when each point is at its maximum displacement.
(i) Explain what is meant by a node of a stationary wave. [1]
(ii) State the number of antinodes of the wave shown in Fig. 4.2. [1]
(iii) State the phase difference between points W and Z on the string. [1]
(iv) A new stationary wave is now formed on the string. The new wave has a frequency that is half of the frequency of the wave shown in Fig. 4.2. The speed of the wave is unchanged.
On Fig. 4.3, draw a position of the string, for this new wave, when each point is at its maximum displacement. [1]
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One end of a wire is attached to a fixed point. A force $F$ is applied to the wire to cause extension $x$. The variation with $F$ of $x$ is shown in Fig. 5.1.
The wire has a cross-sectional area of $4.1 \times 10^{-7} \text{m}^2$ and is made of metal of Young modulus $1.7 \times 10^{11} \text{Pa}$. Assume that the cross-sectional area of the wire remains constant as the wire extends.
(a) State the name of the law that describes the relationship between $F$ and $x$ shown in Fig. 5.1. [1]
(b) The wire has an extension of $0.48 \text{ mm}$.
Determine:
(i) the stress [2]
(ii) the strain. [2]
(c) The resistivity of the metal of the wire is $3.7 \times 10^{-7} \Omega \text{ m}$.
Determine the change in resistance of the wire when the extension $x$ of the wire changes from $x = 0.48 \text{ mm}$ to $x = 0.60 \text{ mm}$. [3]
(d) A force of greater than $45 \text{N}$ is now applied to the wire.
Describe how it may be checked that the elastic limit of the wire has not been exceeded. [1]
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(a) A battery of electromotive force (e.m.f.) 7.8V and internal resistance $r$ is connected to a filament lamp, as shown in Fig. 6.1.
[Image_1: Schematic diagram with 7.8V battery, $r$, and lamp]
Fig. 6.1
A total charge of 750 C moves through the battery in a time interval of 1500 s. During this time the filament lamp dissipates 5.7 kJ of energy. The e.m.f. of the battery remains constant.
(i) Explain, in terms of energy and without a calculation, why the potential difference across the lamp must be less than the e.m.f. of the battery.
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(ii) Calculate:
1. the current in the circuit
current = ........................................................ A [2]
2. the potential difference across the lamp
potential difference = ............................................... V [2]
3. the internal resistance of the battery.
internal resistance = ................................................... $\Omega$ [2]
(b) A student is provided with three resistors of resistances 90$\Omega$, 45$\Omega$ and 20$\Omega$.
(i) Sketch a circuit diagram showing how two of these three resistors may be connected together to give a combined resistance of 30$\Omega$ between the terminals shown. Label the values of the resistances on your diagram.
[Image_2: Terminals diagram] [1]
(ii) A potential divider circuit is produced by connecting the three resistors to a battery of e.m.f. 9.0V and negligible internal resistance. The potential divider circuit provides an output potential difference $V_{OUT}$ of 3.6V. The circuit diagram is shown in Fig. 6.2.
[Image_3: Potential divider diagram with 9.0V battery]
Fig. 6.2
On Fig. 6.2, label the resistances of all three resistors and the potential difference $V_{OUT}$. [2]
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(a) A nucleus of an element X decays by emitting a $\beta^+$ particle to produce a nucleus of potassium-39 $^{39}_{19}\text{K}$ and a neutrino. The decay is represented by $$^{Q}_{S}\text{X} \rightarrow ^{39}_{19}\text{K} + ^{P}_{R}\beta^+ + ^0_0\nu.$$ (i) State the number represented by each of the following letters.
P ...............................
Q ...............................
R ...............................
S ............................... [2]
(ii) State the name of the interaction (force) that gives rise to $\beta^+$ decay.
............................................................................................................................... [1]
(b) A hadron is composed of three identical quarks and has a charge of $+2e$, where $e$ is the elementary charge.
Determine a possible type (flavour) of the quarks.
Explain your working.
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