Mathematics 9709/41 — May/June 2017

Walkthrough

The particle is dropped, so its initial kinetic energy is zero. As it falls, gravitational potential energy is converted into kinetic energy, but some energy is also used to overcome air resistance.

First find the loss in gravitational potential energy:

$$\begin{aligned} \text{PE loss} &= mgh \\ &= 0.6 \times 10 \times 8 \\ &= 48 \text{ J} \end{aligned}$$

Next find the kinetic energy just before the particle hits the ground:

$$\begin{aligned} \text{KE gain} &= \frac{1}{2}mv^2 \\ &= \frac{1}{2} \times 0.6 \times 10^2 \\ &= 30 \text{ J} \end{aligned}$$

If there were no air resistance, these two values would be equal. Here the kinetic energy is smaller than the gravitational potential energy lost, so the missing energy has been transferred to the air as work done against resistance. Therefore:

$$\text{WD against resistance} = 48 - 30 = 18 \text{ J}$$

The work done against air resistance is positive because energy is being removed from the particle's mechanical energy.

Key Takeaways

This question tests the work-energy principle in the presence of a non-conservative force. The total mechanical energy is not conserved because air resistance does negative work on the particle. The loss of gravitational potential energy is split between the gain in kinetic energy and the work done against resistance.

Common Mistakes

• Forgetting that the particle is dropped, so its initial kinetic energy is zero.
• Using the final speed as the speed throughout the fall when calculating kinetic energy.
• Subtracting in the wrong order; the work done against resistance is PE loss minus KE gain.
• Using $g = 9.8$ when the mark scheme expects $g = 10$.

Things to Be Careful About

• Take $g = 10 \text{ m s}^{-2}$ unless the question states otherwise.
• The speed given is the speed just before hitting the ground, so it is the final speed used in the kinetic energy calculation.
• The work done against air resistance is the energy lost from the particle, so it is positive and equal to the difference between the potential energy lost and the kinetic energy gained.
• Always include units: the final answer is in joules.