Physics

(d) Watt – Momentum

Watt is the unit of power, not momentum. Momentum is measured in kg m/s (or N s). The other three pairs are correct.

(b) Multiplying a vector by a scalar

This gives a new vector in the same (or opposite) direction with scaled magnitude. A vector cannot be added to a scalar, and one vector cannot be divided by another.

(c) 50 km/h

Total distance $= 60\times 2 + 30\times 1 = 150$ km and total time $= 3$ h.

$v_{avg} = \frac{150}{3} = 50 \text{ km/h}$

(c) 110 m

Let the fall time be $t$, so $h = \tfrac12 g t^2 = 5t^2$ and the sound travel time is $h/340$. Then $t + \dfrac{5t^2}{340} = 5$, which gives

$t^2 + 68t - 340 = 0 \Rightarrow t \approx 4.68 \text{ s}$

$h = 5(4.68)^2 \approx 109 \text{ m} \approx 110 \text{ m}$

(d) 45 m

At the top $v=0$, so

$H = \frac{u^2}{2g} = \frac{30^2}{2\times 10} = 45 \text{ m}$

(a) Force

The first law says a body stays at rest or in uniform motion unless acted on by an external force, so it defines force qualitatively (the second law measures it).

(b) 10 m

Initial KE $= \tfrac12(1)(20)^2 = 200$ J. When KE is half, 100 J has been converted to potential energy:

$mgh = 100 \Rightarrow h = \frac{100}{1\times10} = 10 \text{ m}$

(b) 30 m

The pressure due to water alone is $4\times10^5 - 1\times10^5 = 3\times10^5$ Pa.

$h = \frac{P}{\rho g} = \frac{3\times10^5}{1000\times10} = 30 \text{ m}$

(a) $v_e = \sqrt{\dfrac{2GM}{R}}$

Setting kinetic energy equal to the magnitude of gravitational potential energy, $\tfrac12 m v_e^2 = \dfrac{GMm}{R}$, gives $v_e=\sqrt{2GM/R}$.

(b) K = °C + 273.15 K

The Kelvin scale starts at absolute zero, which is $-273.15$ °C, so $T(\text{K}) = t(^\circ\text{C}) + 273.15$.

(c) Remains constant

During melting the heat supplied goes into latent heat, which breaks intermolecular bonds rather than raising kinetic energy, so the temperature stays at the melting point.

(a) 400 nm

Frequency stays the same while the wavelength reduces by the refractive index:

$\lambda_{glass} = \frac{\lambda}{\mu} = \frac{600}{1.5} = 400 \text{ nm}$

(b) 5.5 V

$I = \frac{E}{R + r} = \frac{6}{5.5 + 0.5} = 1 \text{ A}$

$V = E - Ir = 6 - 1(0.5) = 5.5 \text{ V}$

(d) 332 m/s

The speed of sound in air at 0 °C is about 332 m/s, and it increases by about 0.6 m/s for each degree Celsius rise.

(b) Proxima Centauri

Proxima Centauri, about 4.24 light years away, is the nearest star to Earth after the Sun. It is slightly closer than Alpha Centauri A and B.

(b) Momentum

Impulse $= F\Delta t = \Delta p$, so its unit is N s, which is the same as kg m/s, the unit of momentum.

(a) Zero

A vector along the x-axis has no projection on the y-axis, so $A_y = A\cos 90^\circ = 0$.

(d) 0 m/s

After one full round the particle returns to its starting point, so the displacement is zero.

$v_{avg} = \frac{\text{displacement}}{\text{time}} = \frac{0}{60} = 0 \text{ m/s}$

(a) 16 s

For opposite directions the relative speed is $36 + 54 = 90$ km/h $= 90 \times \frac{5}{18} = 25$ m/s.

Total distance to cross $= 150 + 250 = 400$ m, so

$t = \frac{400}{25} = 16 \text{ s}$

(d) 1 Newton

By definition, 1 N is the force that gives a mass of 1 kg an acceleration of 1 m/s², since $F = ma$.

(c) Newton's Third Law

The rocket pushes the gases backward (action) and the gases push the rocket forward (reaction), equal and opposite.

(a) 2 m

At maximum height all the kinetic energy becomes potential energy: $mgh = 40$ J.

$h = \frac{40}{2 \times 10} = 2 \text{ m}$

(b) Weight of liquid displaced

By Archimedes' principle, the apparent loss in weight of a body immersed in a liquid equals the weight of the liquid it displaces.

(d) Same as before

$g = \dfrac{GM}{R^2}$, so

$g' = \frac{G(4M)}{(2R)^2} = \frac{4}{4}\cdot\frac{GM}{R^2} = g$

(c) 25°C

$C = \frac{5}{9}(F - 32) = \frac{5}{9}(77 - 32) = \frac{5}{9} \times 45 = 25^\circ\text{C}$

(a) Heat up and cool down quickly

Since $Q = mc\Delta T$, a small $c$ means a small amount of heat causes a large temperature change, so the substance heats up and cools down quickly.

(a) 22°

By Snell's law, $\mu = \dfrac{\sin i}{\sin r}$, so

$\sin r = \frac{\sin 30^\circ}{1.33} = \frac{0.5}{1.33} \approx 0.376 \Rightarrow r \approx 22^\circ$

(b) $E = V + Ir$

The EMF equals the terminal potential difference plus the voltage lost across the internal resistance (lost volts $= Ir$) when current $I$ flows.

(b) Remains same

The speed of sound depends on the medium (and its temperature), not on the source frequency. Doubling the frequency halves the wavelength, since $v = f\lambda$ stays constant.

(d) Spiral

The Milky Way is a barred spiral galaxy with a central bulge and spiral arms.

(b) Watt and Joule per second

Power is energy per time, so $1\ \text{W} = 1\ \text{J/s}$: both are units of power. Newton (force), Joule (energy), Pascal (pressure), Coulomb (charge) and Watt (power) are all different quantities.

(a) 0°

$\vec A\cdot\vec B = AB\cos\theta$

This is greatest when $\cos\theta = 1$, i.e. $\theta = 0^\circ$.

(d) 1 hour

The cars approach each other, so the relative speed is $40+20 = 60$ km/h.

$t = \frac{60}{60} = 1\ \text{h}$

(b) 60 km/h

To pass the platform the train covers its own length plus the platform: $200+300 = 500$ m in 30 s.

$v = \frac{500}{30} = 16.67\ \text{m/s} = 16.67\times 3.6 = 60\ \text{km/h}$

(b) 3000 W

$P = Fv = 600\times 5 = 3000\ \text{W}$

(c) Both forces were equal

Starting from rest, $p = Ft$. Same momentum in the same time means $F = p/t$ is the same for both, whatever the masses.

(c) 13.5 N

Deceleration: $v^2 = u^2 - 2as \Rightarrow 0 = 36 - 2a(4) \Rightarrow a = 4.5\ \text{m/s}^2$.

$F = ma = 3\times 4.5 = 13.5\ \text{N}$

Check with work-energy: $\tfrac12(3)(6)^2 = 54\ \text{J} = F\times 4 \Rightarrow F = 13.5\ \text{N}$.

(d) 150 N

Buoyant force = weight in air − apparent weight in water $= 800 - 650 = 150$ N.

(c) 4g

Since $g=\dfrac{GM}{R^2}$, with $R\to R/2$ and $M$ unchanged, $g'=\dfrac{GM}{(R/2)^2}=4g$.

(b) Half

From $Q=mC\Delta T$, $\Delta T=\dfrac{Q}{mC}$. With the same $Q$ and $m$, $\Delta T\propto 1/C$, so $\dfrac{\Delta T_A}{\Delta T_B}=\dfrac{C_B}{C_A}=\dfrac12$.

(a) 50 J/kg°C

$c=\frac{Q}{m\Delta T}=\frac{5000}{10\times10}=50\ \text{J/kg°C}$

(d) From denser to rarer medium

Total internal reflection needs light to go from a denser to a rarer medium, with the angle of incidence greater than the critical angle.

(b) 1.5V

$V=E-Ir=2-0.5\times1=1.5\ \text{V}$

(a) 1 s

Speed of sound in air at room temperature is about $340$ m/s, so $t=\dfrac{340}{340}=1$ s.

(c) Distant galaxies and stars

Hubble orbits above the atmosphere and gives clear images of distant stars, galaxies and nebulae.

(c) Volt

The seven SI base units are metre, kilogram, second, ampere, kelvin, mole and candela. The volt is a derived unit ($\mathrm{kg\,m^2\,s^{-3}\,A^{-1}}$).

(d) Zero

$|\vec A\times\vec B| = AB\sin\theta$. For parallel vectors $\theta = 0^\circ$, so $\sin\theta = 0$ and the cross product is zero.

(d) None of these

The person returns to the starting point, so the net displacement is zero and the average velocity is $0$.

(The value 96 km/h is the average speed: $\dfrac{2\times80\times120}{80+120}=96$ km/h, not the average velocity.)

(b) 2 s

Take upward as positive. Ball P: $y_P = 40t - 5t^2$. Ball Q (dropped from 80 m): $y_Q = 80 - 5t^2$.

$40t - 5t^2 = 80 - 5t^2 \Rightarrow t = 2\ \text{s}$

(b) 300 N

$a=\frac{30-0}{2}=15\ \text{m/s}^2,\qquad F=ma=20\times15=300\ \text{N}$

(c) Remains same

$a=\dfrac{F}{m}=\dfrac{2F}{2m}=\dfrac{F}{m}$, so the acceleration is unchanged.

(a) 500 J

$W=mgh=5\times10\times10=500\ \text{J}$

(c) $10^5$ Pa

Atmospheric pressure at sea level is $1.013\times10^5$ Pa, which is approximately $10^5$ Pa.

(d) $\frac{g R^2}{G}$

At the surface, $g=\dfrac{GM}{R^2}$, so

$M=\frac{gR^2}{G}$

(a) 27 K and 48.6° F

A kelvin has the same size as a degree Celsius, so the difference is 27 K. A Fahrenheit degree is $\tfrac59$ of a Celsius degree, so

$\Delta T_F=27\times\frac95=48.6^\circ\text{F}$

(c) It is different for different substances

Specific heat capacity $c=\dfrac{Q}{m\Delta T}$ is a characteristic property of a material, with unit $\mathrm{J\,kg^{-1}K^{-1}}$. The heat needed to change state is latent heat.

(c) $2 \times 10^8$ m/s

$v=\frac{c}{n}=\frac{3\times10^8}{1.5}=2\times10^8\ \text{m/s}$

(d) 2 Ω

Each part has resistance $\dfrac{50}{5}=10\ \Omega$. Five equal resistors in parallel give

$R_{eff}=\frac{10}{5}=2\ \Omega$

(a) 0.66 m

$\lambda=\frac{v}{f}=\frac{340}{512}\approx0.66\ \text{m}$

(b) Andromeda Galaxy

Of the galaxies listed, Andromeda (about 2.5 million light years away) is the nearest to the Milky Way; Triangulum, Whirlpool and Sombrero are farther away.

(b) $6\sqrt{5}$ m/sec

The energy conserved is $mgh$ turning into $\tfrac12 mv^2$, so $v=\sqrt{2gh}$. With $g=10\ \text{m/s}^2$ and $h=9$ m, $v=\sqrt{2\times10\times9}=\sqrt{180}=6\sqrt5$ m/s, the only option consistent with this ($mgh=180$ J). Taking the quoted 36 J literally would give $v=6$ m/s, which is not among the options.

(d) Plane area

Time, speed and mass are scalars. A plane area is treated as a vector, since it has magnitude and a direction (along the normal to the surface).

(c) 10 J

By the work-energy theorem, the work done equals the change in kinetic energy: $W = 0 - 10 = -10$ J, so 10 J of work must be done against the motion to stop it.

(a) 4 N

$F=ma$: for the 2 kg mass, $F=2\times2=4$ N. Check with the 4 kg mass: $4\times1=4$ N.

(a) A magnet cannot have two opposite poles

Every magnet has a north and a south pole, so this statement is false. Magnetic monopoles do not exist, so a magnet cannot have an isolated pole either, which means statement (c) is true.

(a) only converge

A convex lens is a converging lens: a parallel beam is brought to a real focus on the other side.

(a) A longitudinal wave.

Particles of the medium vibrate parallel to the direction of travel, forming compressions and rarefactions. Sound needs a medium and is not electromagnetic.

(b) Potential difference

Potential difference is defined as the work done per unit positive charge in moving it between two points, $V=W/q$.

(d) All of the above

Conductivity is $\sigma = 1/\rho$, so it depends on resistivity, which in turn depends on the nature of the material and on temperature.

(b) infinity

The far point of a normal eye is at infinity and the near point is 25 cm.

(c) 20 Hz

The audible range of human hearing is about 20 Hz to 20 kHz.

(d) High specific heat capacity

A liquid with a high specific heat capacity absorbs a large amount of heat for a small rise in temperature, so it cools the body effectively (as water does).

(b) $4^\circ$C

Water has its maximum density at $4^\circ$C, so its volume is minimum there.

(d) 24 hours

The earth completes one rotation about its axis in about 24 hours (one day).

(d) Both a and c

A body becomes negatively charged by gaining electrons, so its number of electrons increases and, since electrons have mass, its mass increases slightly.

(c) It emits energy while moving in orbits

In Bohr's model an electron does not radiate energy while in a stationary orbit; it emits or absorbs energy only when it jumps between orbits. The other statements are true.