Figure 2 shows a moon of mass m in a circular orbit of radius r around a planet of mass M, where m << M - AQA - A-Level Physics - Question 2 - 2020 - Paper 2

Using the data provided for Umbriel, we can substitute T:

We know T for Umbriel is 4.14 days. First, convert this to seconds:

$T = 4.14 \times 24 \times 3600 = 358.56 \text{ s}$

Now substituting the value of T into the equation:

$X = \left(\frac{T^2}{k} \right)^{1/3}$

Substituting k from the previous part:

$X = \left(\frac{(358.56)^2}{\frac{4 \pi^2}{GM}} \right)^{1/3}$

Calculating gives:

$X = 2.65 \times 10^8 m$.

Using the value of X from the previous step:

Calculate M using:

$M = \frac{4 \pi^2 X^3}{T^2}$

Substituting X = 2.65 \times 10^8 m and T = 358.56 s gives:

$M = \frac{4 \pi^2 (2.65 \times 10^8)^3}{(358.56)^2}$

Calculating this we find:

$M \approx 8.56 \times 10^{25} kg$.

The escape velocity ( v_e ) can be calculated using:

$v_e = \sqrt{\frac{2GM}{r}}$

First, consider the masses and diameters:

• For Ariel: ( r = \frac{1.16 \times 10^3}{2} = 0.58 \times 10^3 \text{ m} )
• For Oberon: ( r = \frac{2.52 \times 10^6}{2} = 1.26 \times 10^6 \text{ m} )
• For Titania: ( r = \frac{1.58 \times 10^6}{2} = 0.79 \times 10^6 \text{ m} )

Using these values:

• Ariel: ( v_e \approx \sqrt{\frac{2 \cdot G \cdot 1.27 \times 10^3}{0.58 \times 10^3}} )
• Oberon: ( v_e \approx \sqrt{\frac{2 \cdot G \cdot 3.03 \times 10^3}{1.26 \times 10^6}} )
• Titania: ( v_e \approx \sqrt{\frac{2 \cdot G \cdot 3.49 \times 10^3}{0.79 \times 10^6}} )

Calculating these will show that Titania has the highest escape velocity due to its mass-to-radius ratio.

To determine this, we use the potential energy formula:

$PE = mgh$

Where:

• m is the mass of the object,
• g is the acceleration due to gravity on Ariel,
• h is the height.

At a height of 100 m, we put:

$100 = \frac{PE}{m g}$

By determining g for Ariel and comparing if PE can reach 100 m, you may find that it cannot given Ariel's acceleration and gravitational pull. Therefore, it is unlikely the apparatus could project to a height greater than 100 m.