A-Level Physics May/June 2024 Q4(b)(ii): The ball leaves the spring when the spring reaches its original length. Assume that all…

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  According to the principle of conservation of energy, the initial elastic potential energy ($E_{EPE}$) is converted into the final kinetic energy ($E_K$) of the ball and the gravitational potential energy ($E_{GPE}$) it gains.

  The net energy transferred to kinetic energy is therefore the initial elastic potential energy minus the gain in gravitational potential energy.

  $E_K = E_{EPE} - E_{GPE}$

  The values for energy (from previous parts of the question) and mass are: $E_{EPE} = 9.3 \times 10^{-2} \, \text{J}$ $E_{GPE} = 9.1 \times 10^{-3} \, \text{J}$ $m = 4.5 \times 10^{-2} \, \text{kg}$

  The formula for kinetic energy is: $E_K = \frac{1}{2}mv^2$

  By equating the energy transferred to the kinetic energy gained: $(9.3 \times 10^{-2} - 9.1 \times 10^{-3}) = \frac{1}{2} \times (4.5 \times 10^{-2}) \times v^2$

  $8.39 \times 10^{-2} = (2.25 \times 10^{-2}) \times v^2$

  Now, we rearrange the equation to solve for the speed, $v$: $v^2 = \frac{2 \times (8.39 \times 10^{-2})}{4.5 \times 10^{-2}}$

  $v^2 = 3.7288...$

  $v = \sqrt{3.7288...}$

  $v = 1.9309... \, \text{ms}^{-1}$

  Rounding to two significant figures gives:

  speed = $1.9 \, \text{ms}^{-1}$

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  How the marks are awarded

  • C1 — Correctly stating or using the formula for kinetic energy, $E_K = \frac{1}{2}mv^2$.
  • C1 — Correctly substituting values into the energy conservation equation. This involves equating the net energy available for kinetic energy (initial EPE minus GPE gained) to the kinetic energy formula.
  • A1 — Correctly calculating the final answer for the speed as 1.9 ms⁻¹, or an unrounded value such as 1.93 ms⁻¹.

  Common mistakes

  • Forgetting to subtract the gravitational potential energy gained ($9.1 \times 10^{-3}$ J) from the initial elastic potential energy, and instead incorrectly equating the total initial EPE to the final KE.
  • Making an algebraic error when rearranging the kinetic energy formula, such as forgetting to multiply by 2 or forgetting to take the square root of v² at the final step.
  • Using an incorrect value for the energy transferred, for example by adding the EPE and GPE values instead of subtracting them, indicating a misunderstanding of the energy transformations.
  • Using a value for mass in grams (e.g., 45 g) instead of kilograms (0.045 kg or $4.5 \times 10^{-2}$ kg), leading to an incorrect final answer.

  Examiner tip: For any energy conservation problem, always start by writing down an energy balance equation that accounts for all initial and final energy forms involved in the process.

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