Show that the intensity of the solar radiation at the location of Titan is $16 W m^{-2}$.

Titan has an atmosphere of nitrogen. The albedo of the atmosphere is 0.22. The surface of Titan may be assumed to be a black body. Explain why the average intensity of solar radiation absorbed by the whole surface of Titan is $3.1 W m^{-2}$.

Show that the equilibrium surface temperature of Titan is about $90 K$.

The mass of Titan is 0.025 times the mass of the Earth and its radius is 0.404 times the radius of the Earth. The escape speed from Earth is 11.2 km/s$^{-1}$. Show that the escape speed from Titan is 2.8 km/s$^{-1}$.

The orbital radius of Titan around Saturn is $R$ and the period of revolution is $T$.

Show that $T^2 = \frac{4\pi^2 R^3}{GM}$ where $M$ is the mass of Saturn.

The orbital radius of Titan around Saturn is $1.2 \times 10^9$ m and the orbital period is 15.9 days. Estimate the mass of Saturn.

Show that the mass of a nitrogen molecule is $4.7 \times 10^{-26}$ kg.

Estimate the root mean square speed of nitrogen molecules in the Titan atmosphere. Assume an atmosphere temperature of $90 K$.

Discuss, by reference to the answer in (b), whether it is likely that Titan will lose its atmosphere of nitrogen.

Explain what is meant by the gravitational potential at the surface of a planet.

An unpowered projectile is fired vertically upwards into deep space from the surface of planet Venus. Assume that the gravitational effects of the Sun and the other planets are negligible.

The following data are available.

• Mass of Venus = $4.87\times10^{24}$ kg
• Radius of Venus = $6.05\times10^{6}$ m
• Mass of projectile = $3.50\times10^{3}$ kg
• Initial speed of projectile = $1.10\times\text{escape speed}$

(i) Determine the initial kinetic energy of the projectile.

(ii) Describe the subsequent motion of the projectile until it is effectively beyond the gravitational field of Venus.

Outline what is meant by electric potential at a point.

The electric potential at a point a distance $2.8\ \text{m}$ from the centre of the sphere is $7.71\ \text{kV}$. Determine the radius of the sphere.

Comment on the angle at which the object meets equipotential surfaces aroundthe sphere.

Determine whether the object will reach the surface of the sphere.

Show that the kinetic energy of the object is about $0.7\, \text{mJ}$.

Outline how this diagram shows that the gravitational field strength of planet X decreases with distance from the surface.

The diagram shows part of the surface of planet X. The gravitational potential at the surface of planet X is $-3V$ and the gravitational potential at point Y is $-V$.

Sketch on the grid the equipotential surface corresponding to a gravitational potential of $-2V$.

A meteorite, very far from planet X begins to fall to the surface with a negligibly small initial speed. The mass of planet X is $3.1 \times 10^{21}$ kg and its radius is $1.2 \times 10^{6}$ m. The planet has no atmosphere. Calculate the speed at which the meteorite will hit the surface.

Show that the energy of photons from the UV lamp is about $10 \text{ eV}$.

Calculate, in J, the maximum kinetic energy of the emitted electrons.

Suggest, with reference to conservation of energy, how the variable voltage source can be used to stop all emitted electrons from reaching the collecting plate.

The variable voltage can be adjusted so that no electrons reach the collecting plate. Write down the minimum value of the voltage for which no electrons reach the collecting plate.

On the diagram, draw and label the equipotential lines at $-0.4$ V and $-0.8$ V.

An electron is emitted from the photoelectric surface with kinetic energy $2.1$ eV. Calculate the speed of the electron at the collecting plate.

Show that the speed ${v}$ of the electron with mass ${m}$, is given by ${v}=\sqrt{\frac{2{k}{e}^{2}}{mr}}$ a(i).

Hence, deduce that the total energy of the electron is given by ${E_{TOT} = -\frac{k \cdot e^2}{r}}$.

a(ii).

In this model the electron loses energy by emitting electromagnetic waves. Describe the predicted effect of this emission on the orbital radius of the electron. a(iii).

Show that the de Broglie wavelength $\lambda$ of the electron in the $n=3$ state is $\lambda=5.1\times10^{-10}$ m.

The formula for the de Broglie wavelength of a particle is $\lambda=\frac{h}{mv}$. b(i).

Estimate for $n=3$, the ratio $\frac{\text{circumference of orbit}}{\text{de Broglie wavelength of electron}}$.

State your answer to one significant figure.

b(ii).

The description of the electron is different in the Schrodinger theory than in the Bohr model. Compare and contrast the description of the electron according to the Bohr model and to the Schrodinger theory.