Bouncing ball

Study energy conservation during ball bounces by measuring the time intervals between each impact using the smartphone microphone.

When you drop a ball on the floor, it bounces but never reaches back to its starting height. With each bounce, some kinetic energy is lost: it transforms into heat, sound, and permanent deformation. But how much energy is lost at each bounce? By recording the sound of the impacts with the smartphone microphone and measuring the time intervals between bounces, you can calculate the rebound height and determine the coefficient of restitution — all without needing a ruler or camera.

Visão geral da atividade:

The student drops a ball on a hard surface next to the smartphone recording sound level. Each bounce produces a clearly identifiable sound peak. By measuring the time intervals between bounces, the student calculates the rebound height and determines the coefficient of restitution.

Nível:

High School

FizziQ

Autor:

Duração (minutos):

25

O que os alunos farão:

- Measure the time intervals between ball bounces using the microphone
- Calculate the rebound height from the time intervals using free-fall equations
- Determine the coefficient of restitution from the ratio of successive heights
- Verify that the coefficient of restitution remains approximately constant
- Compare different ball materials and their energy dissipation properties

Conceitos científicos:

- Inelastic collision
- Coefficient of restitution
- Energy conservation
- Kinetic and potential energy
- Free fall
- Energy dissipation

Sensores:

- Microphone (sound level / amplitude)

Materiais necessários:

- Smartphone or tablet with FizziQ
- Small dense ball (steel ball, squash ball, bouncing ball)
- Hard flat surface (tile, concrete, table)

Procedimento experimental:

Open FizziQ and select the Sound Level (amplitude) instrument. Place the smartphone flat on a hard surface.
Prepare a small dense ball (steel ball, squash ball, bouncing ball). Balls that produce a sharp impact sound work best.
Start the sound recording in FizziQ.
Drop the ball from a known height (for example 50 cm) right next to the smartphone microphone, without throwing it — just let it fall.
Let the ball bounce freely until it stops (typically 5 to 8 audible bounces).
Stop the recording and observe the sound level graph as a function of time.
Identify the peaks corresponding to each impact. Note the times t₁, t₂, t₃... of each bounce.
Calculate the intervals Δtₙ = tₙ₊₁ - tₙ between two successive bounces. The maximum height between two bounces is: hₙ = g × (Δtₙ/2)² / 2.
Calculate the energy ratio conserved after each bounce: Eₙ₊₁/Eₙ = hₙ₊₁/hₙ = (Δtₙ₊₁/Δtₙ)². This ratio is the square of the coefficient of restitution e.
Verify that the coefficient of restitution remains approximately constant from one bounce to the next. Compare it for different ball types.

Resultados esperados:

The sound graph shows intensity peaks that become increasingly close together, corresponding to successive bounces. The intervals between bounces form a geometric sequence with ratio e (the coefficient of restitution). Typical values: steel ball e ≈ 0.90-0.95, rubber ball e ≈ 0.80-0.85, tennis ball e ≈ 0.70-0.75. The coefficient remains approximately constant across bounces (within ±5%).

Questões científicas:

- Why do the intervals between bounces form a geometric sequence?
- What happens if you drop the ball from a greater height: does the coefficient of restitution change?
- Why does a squash ball bounce less than a tennis ball?
- What fraction of kinetic energy is lost at each bounce for a ball with e = 0.85?
- Why does the ball eventually stop even though each bounce conserves a fixed fraction of energy?
- How could you determine the drop height from only the sound recording?

Explicações científicas:

In an inelastic collision, kinetic energy is not conserved (unlike a perfectly elastic collision). The coefficient of restitution e = v_after/v_before quantifies how much velocity is retained after impact.

For a ball in free fall, the velocity just before a bounce equals v = g × Δt/2, where Δt is the time interval between two successive bounces. This formula comes from the symmetry of free fall: the ball takes half the interval to go up and half to come back down.

The coefficient of restitution depends on the ball material and the surface. Typical values: tennis ball ≈ 0.75, rubber ball ≈ 0.80-0.90, steel ball on steel ≈ 0.95, squash ball ≈ 0.40.

The energy lost at each bounce dissipates as heat (material deformation), sound (the impact we hear), and sometimes light (sparks with steel). For a rubber ball, heat from internal friction dominates.

For a modeling clay ball, the collision is nearly perfectly inelastic (e ≈ 0): all kinetic energy is absorbed by permanent deformation and the ball does not bounce.

The height reached between two bounces is calculated by hₙ = ½g(Δtₙ/2)², considering that the ball performs a symmetric free-fall trajectory between two impacts.

The ratio of successive energies equals e² because energy is proportional to the square of velocity: Eₙ₊₁/Eₙ = (vₙ₊₁/vₙ)² = e².

The total energy dissipated after n bounces equals E₀ × (1 - e^(2n)), which explains why the ball always eventually stops.

Atividades de extensão:

Perguntas frequentes:

Q: Some bounces are not detected by the microphone.
R: Increase the detection threshold sensitivity. If the ball is small and light, bring it closer to the microphone. Use a denser ball for clearer impacts.

Q: The coefficient of restitution varies a lot between bounces.
R: The first bounce may differ because the ball may not fall perfectly vertically. Ignore the first bounce and use bounces 2-6 for the most consistent results.

Q: Can I use a basketball or football?
R: These are too large and light for precise measurements. Use a small, dense ball that produces sharp, distinct impact sounds.

Q: How do I know the initial drop height if I did not measure it?
R: You can calculate it from the first time interval: h₀ = g(Δt₁)²/8, but this assumes the first impact occurs at t = 0.