Energy Calculations: KE, GPE and Conservation

1. Definition

Energy calculations using conservation

Energy calculations use the principle of conservation of energy to relate changes in energy stores (especially GPE and KE) and any energy dissipated to the surroundings.

2. Key Ideas

• Core equations (O Level):
  • kinetic energy: Eₖ = (1/2)mv²
  • change in gravitational potential energy near Earth: Δ Eₚ = mgΔ h
• Eₖ and Δ Eₚ are in joules (J); m is mass in kilograms (kg), v is speed in m s⁻¹, g is gravitational field strength in N kg⁻¹, and Δ h is the vertical height change in metres (m). The numerical value of g may also be written in m s⁻².
• Start by identifying the initial and final energy stores.
• Define the system and reference level when they matter. For GPE calculations, only Δ h matters; the zero level may be chosen for convenience.
• For the common KE–GPE model, write: initial GPE + initial KE = final GPE + final KE + energy dissipated.
• State what is neglected. For example, “air resistance is negligible” means there is no dissipated-energy term for the flight.
• Keep units consistent and convert early: g → kg and cm → m.
• Use the value of g given in the question.

3. Detailed Explanations

A quick method for energy questions

1. Mark the initial and final states, including the vertical height change.
2. List the energy stores at each state and any energy dissipated.
3. Write the conservation equation in words before using formulas.
4. Substitute SI values, solve, and check the unit and size of the answer.

For example, if an object falls from rest through vertical height h with negligible air resistance, mgh = (1/2)mv² ⇒ v = square root of 2gh

The mass cancels because, under these assumptions, both the GPE decrease and KE increase are proportional to mass.

Case study 1: Oscillating pendulum

Choose the lowest point B as the zero-GPE reference. At the turning point A, the bob is momentarily at rest, so its KE is zero and its GPE is maximum. With negligible resistance, the GPE decrease from A to B equals the KE increase.

For an ideal pendulum with negligible air resistance and negligible friction at the pivot:

• A → B: GPE decrease = KE increase
• B → D: KE decrease = GPE increase

In equation form:

From point A to point B,

E_(p,A) = E_(k,B)

From point B to point D,

E_(k,B) = E_(p,D)

For a real pendulum, air resistance and friction transfer energy from the pendulum’s mechanical stores to internal energy stores of the pendulum and surroundings. Some energy is also transferred away by sound. The next turning point is therefore lower, although total energy is conserved.

Across the real swing from A to the lower turning point D′,

E_(p,A) = E_(p,D') + E_(dissipated, A to D')

Case study 2: Bouncing ball

Consider a ball of mass m dropped from height hᵢₙᵢₜᵢₐₗ. Choose the ground as the zero-GPE reference and neglect air resistance during each flight.

Stage 1: Initially, the ball has:

• GPE is mghᵢₙᵢₜᵢₐₗ.
• KE is zero because the ball starts from rest.

Stage 2: Just before the ball touches the ground, the ball has:

• GPE is approximately zero at the chosen reference level.
• KE is mghᵢₙᵢₜᵢₐₗ because the GPE decrease equals the KE increase.

Stage 3: When the ball hits the ground and rebounds:

• The ball briefly deforms, storing energy elastically before regaining its shape.
• Some energy is transferred to internal energy stores of the ball and ground, and some is carried away by sound waves.
• The KE just after rebound is therefore smaller than the KE just before impact.

Stage 4: The ball rises to a smaller maximum height, hₘₐₓ:

• GPE is mghₘₐₓ and KE is zero.
• Because energy was dissipated during the collision, hₘₐₓ < hᵢₙᵢₜᵢₐₗ.

If a question includes air resistance, add energy dissipated during the flight as well as during the collision.

4. Common Mistakes

• Using g = 10 without checking what the question states.
• Forgetting unit conversions ( g → kg and cm → m).
• Forgetting to include an energy dissipated term when friction/air resistance is mentioned.
• Using the distance along a slope as h (for GPE, h is the vertical height change).
• Writing “energy is lost” as if it has been destroyed. Instead, state the store that decreases and where energy is transferred.

5. Exam Tips

• Start with a one-line equation: initial GPE + initial KE = final GPE + final KE + energy dissipated. Remove any zero terms only after writing the full model.
• For an object dropped from rest, with the ground as the zero-GPE reference and negligible air resistance, its energy just before impact is all KE.
• If resistive forces are negligible, decrease in GPE = increase in KE is often the fastest route.
• If the object is at rest (e.g. at maximum height), set Eₖ = 0.
• Always give a final line with a unit (J for energy, m s⁻¹ for speed).

6. Worked Examples

7. Mind Stretchers