describe an experiment to determine the acceleration of free fall using a falling object

Apparatus

Method

1. Set‑up
   • Secure the retort stand on a stable bench.
   • Mount the photogate(s) so that the infrared beams are vertical and centred on the falling path.
   • Measure the vertical distance s from the centre of the release point to the centre of the first photogate. Record Δs (± 0.001 m).
   • If two gates are used, measure the separation d between them (± 0.001 m).
2. Calibration (optional but recommended) – Verify the photogate timer with a known interval (e.g., a calibrated electronic pulse generator) and note any systematic offset.
3. Perform the fall
   • Attach the ball to the release mechanism at the measured height.
   • Release the ball smoothly so that the initial speed \(u = 0\).
   • For a single gate the timer starts when the ball interrupts the beam and stops when it clears the same beam; the displayed time is t.
   • For two gates the timer records the interval t₁₂ between the first and second interruptions.
4. Record data – For each height repeat the fall at least 5 times and note:
   • Height s (m) and its uncertainty Δs
   • Time t (s) or interval t₁₂ (s) and uncertainty Δt (± 0.001 s)
   • Calculate \(t^{2}\) (or \(v = d/t_{12}\) and \(v^{2}\)) for later analysis.
5. Vary the height – Suggested heights: 0.40 m, 0.55 m, 0.70 m, 0.85 m, 1.00 m. Record a full data set for each method.
6. Clean‑up – Switch off the release mechanism, dismantle the set‑up and store equipment safely.

Data Collection (example)

Data Analysis

1. Single‑gate method
   • For each height compute \(g = 2s/t^{2}\) and the mean value \(\overline{g}\).
   • Plot **\(s\) (vertical axis) against \(t^{2}\) (horizontal axis)**.
     • The straight‑line equation is \(s = \tfrac12 g\,t^{2}\); the gradient \(m = \tfrac12 g\).
     • Use linear regression (or a spreadsheet) to obtain the best‑fit gradient and its standard error.
     • Calculate \(g = 2m\) and the associated uncertainty.
2. Two‑gate (velocity) method
   • Calculate the average velocity between the gates: \(v = d/t_{12}\).
   • Obtain \(g = v^{2}/(2s)\) for each height and average the results.
   • Plot **\(v^{2}\) (vertical) against \(2s\) (horizontal)** – the gradient should be the experimental value of \(g\).
3. Comparison of methods – Present the two sets of results in a table and comment on any systematic difference.

Uncertainty & Error Analysis

• Propagation of uncertainties (AO2)

  For the single‑gate formula \(g = 2s/t^{2}\):

  \[ \frac{\Delta g}{g}= \sqrt{\left(\frac{\Delta s}{s}\right)^{2} + \left(2\frac{\Delta t}{t}\right)^{2}} \]

  Insert the measured \(\Delta s\) and \(\Delta t\) to obtain \(\Delta g\). The same approach applies to the two‑gate formula \(g = v^{2}/(2s)\) with \(\Delta v = v\,\sqrt{(\Delta d/d)^{2}+(\Delta t_{12}/t_{12})^{2}}\).

• Random errors
  • Timing resolution of the photogate (± 0.001 s).
  • Small variations in the release height (human positioning).
  • Air currents causing slight lateral motion.
• Systematic errors
  • Mis‑alignment of photogates (beam not exactly vertical).
  • Delay in the release mechanism (non‑zero initial speed).
  • Parallax when reading the height on the ruler.
  • Unaccounted air‑resistance – becomes noticeable for larger fall distances.
• Minimisation strategies
  • Use the longest practical fall distance to reduce the relative impact of \(\Delta t\).
  • Check gate alignment with a plumb line; ensure the ball passes through the centre of each beam.
  • Calibrate the release mechanism by measuring the time for a very short drop; adjust if a consistent offset is observed.
  • Repeat each measurement ≥ 5 times and use the mean value.

Link to Other Syllabus Topics (AS‑Level Integration)

Evaluation (Example)

From the example data the mean values are:

• Single‑gate: \(\overline{g}=9.80\;\text{m s}^{-2}\) with \(\Delta g = \pm0.04\;\text{m s}^{-2}\) (≈ 0.4 %).
• Two‑gate: \(\overline{g}=9.50\;\text{m s}^{-2}\) with \(\Delta g = \pm0.06\;\text{m s}^{-2}\).

Both methods agree with the accepted value \(9.81\;\text{m s}^{-2}\) within their uncertainties, although the two‑gate method gives a slightly lower result, likely due to a small systematic under‑estimation of the gate separation d.

Key sources of error

• Timing uncertainty dominates for short falls – longer heights improve precision.
• Gate mis‑alignment introduced a systematic bias in the two‑gate method.
• Air resistance was negligible for the dense metal sphere but would be significant for a lighter object.

Possible improvements

1. Use a high‑resolution timer (0.000 s) to reduce \(\Delta t\).
2. Measure the gate separation d with a micrometer for greater accuracy.
3. Conduct the experiment in a low‑pressure tube to eliminate drag and demonstrate the ideal free‑fall condition.
4. Repeat the trial with objects of different mass and surface area to explore the effect of air resistance.

Safety Considerations

• Clear the area directly beneath the falling object; no person should stand under the apparatus.
• Secure the retort stand and clamps to prevent tipping.
• Use a solid metal sphere – avoid glass or hollow objects that could shatter.
• Handle the electromagnetic release mechanism according to the manufacturer’s instructions; keep fingers clear of moving parts.
• After each series of drops, check that the photogate beams are unobstructed and that the ruler has not shifted.