AQA A-Level Physics Paper 2, June 2023: Question 11

1 mark · Easy difficulty · Multiple Choice

Identify the relationship showing what planetary escape velocity is directly proportional to in terms of mass and radius.

Practise this question

Question

Question 11 asks: Data are collected for the mass M, radius R and escape velocity u for each planet in the Solar System. The data show that u is directly proportional to: option A is (M/R)^(-1/2), option B is (M/R)^(1/2), option C is M/R, and option D is (M/R)^2. Each option has a selection box beside it, with a mark value of 1 mark.
Question text

11 Data are collected for the mass M, radius R and escape velocity u for each planet in the

Solar System.

The data show that u is directly proportional to

[1 mark]

−

M 2

A

R

M 2

B

R

M

C

R

M

D

R

Mark scheme

Show the mark scheme Mark scheme table row showing question 11 with correct answer B, and the corresponding expression (M/R)^(1/2).

11 B M 2

R

How to answer it

Gravitational Fields: Proportionality of Escape Velocity

What this question tests

This question assesses your understanding of escape velocity in a radial gravitational field, specifically your ability to apply the principle of conservation of energy to equate kinetic energy to gravitational potential energy change, and deduce proportionality relationships between velocity, mass, and radius.

Question 11 (1 Mark)

Multiple Choice Analysis

Identifying the correct proportionality for planetary escape velocity u

✅ Correct Answer

Option B: (M / R)1/2

Mark Scheme: 1 mark awarded for selecting B.

💡 Key Knowledge

  • Escape Velocity (u): The minimum initial speed an object needs at a planet's surface to escape its gravitational field to infinity with zero remaining kinetic energy.
  • Conservation of Energy:
    Kinetic Energy supplied at surface = Work done against gravity to reach infinity (Gravitational Potential Energy change).
  • Gravitational potential at the surface of a spherical mass M of radius R is:
    V = -GM / R

📐 Step-by-Step Derivation

  1. Set up energy conservation:
    A projectile of mass m launched at speed u from radius R escaping to infinity (where potential is 0 and speed is 0) satisfies:
    ½ m u² = G M m / R
  2. Cancel the mass of the projectile (m):
    ½ u² = G M / R
  3. Rearrange for escape velocity u:
    u² = 2 G M / R
    u = √(2 G M / R) = (2 G)1/2 × (M / R)1/2
  4. Extract proportionality:
    Since 2 and G are constants:
    u ∝ (M / R)1/2

🧠 Exam Technique

  • Quick check via dimensions: Kinetic energy is proportional to v² , whereas potential energy is proportional to M / R . Therefore, velocity must scale with the square root: (M / R)1/2 .
  • Power notation: Remember that the square root is equivalent to a power of ½ or 0.5 . Negative indices denote division, so (M / R)-1/2 would mean √(R / M) .

❌ Common Distractor Traps

  • Selecting C (M / R): Forgetting to take the square root of both sides after equating ½ m u² with G M m / R .
  • Selecting D (M / R)²: Squaring instead of square-rooting.
  • Selecting A (M / R)-1/2: Confusing the negative sign from the potential formula V = -GM/R and applying it incorrectly into the exponent.

Topics

Physics · 3.7 Fields and their consequences (A-level only)

Question and mark scheme from the AQA A-Level Physics examination, Paper 2, June 2023. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.