Gravitation & GPE Question Untangler
A deeper course for students who understand the basic ideas but lose marks when a question hides the physics inside powers of ten, squared units, radius/altitude wording, or several equations at once.
Foundations: see the relationships first
Before the harder questions, make the four basic pictures automatic. The diagrams use the same centre-to-centre radius and the same zero-at-infinity reference throughout.
1. Radius means from the centre
Altitude starts at the surface. Universal gravitation uses the full centre-to-object distance.
2. Field strength falls faster
Doubling the centre distance makes field strength one quarter as large.
3. Potential falls as 1/r
Potential remains negative for a bound position, but becomes less negative as the object moves away.
4. Zero at infinity
Negative gravitational potential energy is not an error. It records that the object is bound relative to infinity.
Build the gravitation map first
The biggest source of confusion is treating all gravitational equations as if they behave the same way. They do not.
Depends on 1/r². Unit: N.
Depends on 1/r². Unit: N kg⁻¹ = m s⁻².
Depends on 1/r. Unit: J kg⁻¹.
Depends on 1/r. Unit: J.
Powers of ten and squared units
This is where a simple conversion can become a factor-of-a-million error.
The key idea: square the conversion factor too
If 1 km = 10³ m, then:
(1 km)² = (10³ m)² = 10⁶ m²
Therefore 1 km² = 10⁶ m², not 10³ m².
Scientific notation when squaring
(6.40 × 10⁶)²
Square the coefficient: 6.40² = 40.96
Square the power of ten: (10⁶)² = 10¹²
So: 40.96 × 10¹² = 4.096 × 10¹³
6.4e6^2, different calculators/apps may interpret the expression differently. Safest: enter (6.4e6)² with brackets.Why the units of G matter
The m² inside G is telling you something important: if you use G in this SI form, your r must be in metres before you square it. Using kilometres in the denominator while leaving G in N m² kg⁻² creates a hidden factor of 10⁶ error.
| If you see… | Ask yourself… |
|---|---|
| r² in a formula | Have I converted r first, and only then squared it? |
| km² | Did I multiply by 10⁶ to get m²? |
| cm² | Did I multiply by 10⁻⁴ to get m²? |
| (a × 10ⁿ)² | Did I square both a and 10ⁿ? |
| G in SI units | Are r and mass also in SI units? |
Radius, altitude and separation
The symbol r does not mean “whatever distance appears in the question”. It means the separation of the centres of mass.
Three different wordings
A “400 km above the surface” → convert h, then r = R + h.
B “6.8 × 10⁶ m from Earth’s centre” → that number is already r.
C “altitude equal to 2R” → r = R + 2R = 3R.
Gravitational field strength and force
These are the inverse-square quantities. This is also where squared distances and unit consistency matter most.
Worked structure: g at altitude
Earth radius: 6.37 × 10⁶ m, altitude: 400 km.
1. Convert altitude: 400 km = 4.00 × 10⁵ m.
2. Find r: 6.37 × 10⁶ + 4.00 × 10⁵ = 6.77 × 10⁶ m.
3. Square the entire radius: (6.77 × 10⁶)².
4. Substitute into g = GM/r².
Potential and gravitational potential energy
These are inverse-r quantities, not inverse-square quantities.
Changes in GPE: the sign is part of the physics
IB questions often ask for the energy needed to move an object rather than its GPE at one point.
If r₂ > r₁, then 1/r₁ > 1/r₂, so ΔEₚ is positive.
That matches the physical story: to move an object farther away against attraction, an external agent must supply energy.
Mixed medium problems
Now the question will not tell you which difficulty it is testing. You must diagnose it.
Hard IB-style untangling
These deliberately combine wording, geometry, powers, signs and proportional reasoning.
Section 2 — IB Physics HL Practice
Twenty fresh questions designed to make you decode the physics before calculating: 10 medium and 10 hard. These are original IB-style practice questions, not past-paper reproductions.
Medium HL — 10 questions
These combine two or three ideas at a time: proportional reasoning, potential, work, gradients, escape speed and circular orbits.
Hard HL — 10 questions
These require several decisions in sequence. Expect geometry, symbolic reasoning, energy, field/potential links, orbit relations and careful interpretation.
Beyond standard HL — challenge set
These are deliberately harder than the earlier practice. They require symbolic links between equations, changing reference points, orbital energy and careful interpretation of the wording.
Mastery summary
A high score matters less than being able to explain why every radius, exponent and sign is there.
Gravitational Potential — from V to orbital energy
A focused route for the part of gravitation that is easiest to confuse: potential versus potential energy, negative values, potential differences, escape, orbits and the link between V and g.
Build the potential map
Keep the quantity, unit, sign and reference point visible before substituting.
Energy per unit mass. Unit J kg⁻¹. It belongs to a position in a field.
Energy of the two-body system. Unit J. It depends on the test mass.
Potential is a position map
A test mass placed at the same point has the same V, whatever its own mass.
Potential difference becomes energy
Units make the relationship visible: J kg⁻¹ × kg = J.
Potential and potential difference
Potential is a scalar. Add contributions algebraically, and use differences for energy transfers.
Energy, escape and circular orbits
These questions combine potential with kinetic energy and conservation laws.
At infinity, final total energy is zero.
Total energy E = Eₚ/2 = −GMm/(2r).
For a spherical source outside the surface.
Hard HL potential synthesis
These are intentionally beyond routine substitution. Show the symbolic relationship first.
Mastery summary
Explain every sign and every radius aloud before you count a question as secure.
Gravitational potential energy & potential
Work through what each question means, choose the next step, then reach the result. Check your choice or reveal one solution step whenever you want.
Solutions are revealed one step at a time. Moving to another step never reveals its answer.