Part 1: The Physics Behind the Practical

The experiment relies on the SUVAT equation for constant acceleration. When an object falls from rest (or from some initial velocity $u$), the displacement $h$ after time $t$ is given by the standard kinematics relation.

Here $h$ is the vertical distance fallen between the two timing points, $u$ is the velocity of the ball bearing as it passes the first timing point, $g$ is the acceleration due to gravity (the quantity we want), and $t$ is the time to fall the distance $h$.

Why you cannot just use $h = \tfrac{1}{2}gt^{2}$

If the ball were released from rest at exactly the point where timing begins, $u$ would be zero and you could plot $h$ against $t^{2}$ to get a straight line through the origin with gradient $\tfrac{1}{2}g$. In practice, timing begins at the upper light gate, but the ball has already been accelerating since it left the electromagnet above. That means $u \neq 0$, and there is an unknown initial velocity to deal with. We therefore need a linearisation that accounts for $u$.

The $2h/t$ vs $t$ linearisation (full derivation)

Starting from the displacement equation:

Divide both sides by $t$ (valid because $t > 0$ for every data point):

Now multiply both sides by 2:

Compare this with $y = mx + c$. If you plot $\dfrac{2h}{t}$ on the $y$-axis against $t$ on the $x$-axis, you obtain a straight line where the gradient equals $g$ and the $y$-intercept equals $2u$. This is the standard linearisation for the two-light-gate version of the practical.

Notice that the intercept $2u$ is not needed for the final answer but its presence confirms that the ball had a non-zero velocity at the first timing point. If $2u$ came out negative or implausibly large, that would signal a systematic error.

If you need to justify the graph choice, show the algebra explicitly: start from $h = ut + \tfrac{1}{2}gt^{2}$, divide by $t$, multiply by 2, and then compare with $y = mx + c$.

Alternative linearisation: $h$ vs $t^{2}$ (release from rest)

If you use an electromagnet-and-trapdoor setup where timing starts at the moment of release (so $u = 0$), the equation simplifies to $h = \tfrac{1}{2}gt^{2}$. Plotting $h$ against $t^{2}$ gives a straight line through the origin with gradient $\tfrac{1}{2}g$. This is simpler, but it only works when timing begins exactly at release.

Part 2: Equipment, Setup, and Method

Equipment list

Method A: Electromagnet with two light gates (recommended)

The schematic below shows the full apparatus layout; notice the fixed gap from the electromagnet to the upper light gate, the measured separation $h$ between the two light gates, and the plumb-line alignment through both gates.

[DIAGRAM: asset_name: RP 03 - Determination of g by a Free-Fall Method - Diagram 1; asset_slug: RP 03 - Determination of g by a Free-Fall Method - Diagram 1; recommended_method: retained_png; description: Apparatus setup. Tall retort stand clamped to bench. At the top, an electromagnet connected to a low-voltage DC supply holds a steel ball bearing. Below, two light gates are clamped to the stand, separated by a vertical distance $h$. The upper light gate is connected to start the electronic timer; the lower light gate stops it. A metre ruler is clamped vertically alongside the light gates. At the base, a felt pad catches the ball. A plumb line hangs from the electromagnet to verify vertical alignment. Labels: electromagnet, DC supply, steel ball bearing, upper light gate (starts timer), lower light gate (stops timer), height $h$, metre ruler, felt pad, counterweight on base.]

1. Clamp the retort stand securely to the bench using a G-clamp, or place a 2 kg counterweight on the base. The stand must not move during the experiment, because any wobble changes the fall path.

2. Mount the electromagnet at the top of the stand. Connect it to a low-voltage DC supply set at the manufacturer's recommended voltage.

3. Attach the two light gates to the stand using bosses. The upper light gate should be a fixed distance below the electromagnet (around 5 cm). This gap means the ball is already moving when it reaches the upper gate, giving the non-zero initial velocity $u$ discussed in Part 1.

4. Use a plumb line hung from the electromagnet to ensure the ball bearing will fall vertically through the centres of both light gates. Adjust the gates laterally until the plumb line passes through them. This step is essential because if the ball clips the edge of a gate, the timing will be inaccurate and the ball's trajectory will be deflected.

5. Connect the light gates to the electronic timer or data logger. The upper gate triggers the start; the lower gate triggers the stop.

6. Set the initial height $h = 0.500\,\text{m}$ between the upper and lower light gates, measured using the metre ruler. Clamp the ruler vertically alongside the gates and use a set square to read off the positions of the gate beam centres, reducing parallax error.

7. Switch on the electromagnet and hang the ball bearing from it. Reset the timer to zero.

8. Switch off the electromagnet to release the ball. The ball falls freely, triggering the upper gate (starting the timer) and the lower gate (stopping the timer). Record the time $t$.

9. Repeat the drop three times at this height and calculate the mean time. Three repeats allows you to identify anomalies and reduces the effect of random timing errors.

10. Move the lower light gate up by 0.050 m to reduce $h$ to $0.450\,\text{m}$. Repeat the measurement process.

11. Continue reducing $h$ in steps of $0.050\,\text{m}$ down to $h = 0.250\,\text{m}$. Below this distance, the fall times become so short that percentage uncertainties in timing become unacceptably large.

12. Record all data in a table with columns: $h$/m, $t_{1}$/s, $t_{2}$/s, $t_{3}$/s, mean $t$/s, $2h/t$ / m s$^{-1}$.

Method B: Electromagnet with trapdoor/impact switch (alternative)

In this arrangement, the timer is set to start when the release circuit is triggered and stop when the ball strikes a trapdoor or impact switch at the bottom. If the ball is released physically at the same instant the timer starts, then $u = 0$ and the appropriate graph is $h$ against $t^{2}$, giving a straight line through the origin with gradient $\tfrac{1}{2}g$.

The subtle point is the release delay. If the timer starts electrically when the electromagnet is switched off, but the ball remains stuck to the magnet for a short extra time because of residual magnetism, the recorded time is too long. That makes $t^{2}$ too large, the graph gradient too small, and the calculated value of $g$ too low. A mechanical release that starts timing at the instant the ball actually leaves the support deals with this more cleanly. Method A with two light gates usually avoids this problem and gives more precise timing.

AT k: using software and data logging well

AT k is not just "let the computer do the graph". A data logger or spreadsheet can record the light-gate times, calculate mean values, create a processed column such as $2h/t$ or $t^{2}$, and fit a straight line. You still need to choose the correct graph for the method used, check the units, inspect any outliers, and decide whether the intercept should be free or forced.

For Method A, plot $2h/t$ against $t$ and keep the intercept free because it represents $2u$. For Method B, plot $h$ against $t^{2}$; only in that case should the line be expected to pass through the origin.

Part 3: Variables, Controls, and Experimental Design

Identifying variables clearly helps you describe the method properly and judge the quality of the data. Be specific about instruments and methods of control.

Repeats

Each height should be repeated at least three times to allow calculation of a mean time. This reduces the effect of random variation in the ball's release (slight differences in how cleanly it detaches from the electromagnet).

Ensuring a suitable range

The range $0.250\,\text{m}$ to $0.500\,\text{m}$ gives at least six data points. For the most reliable gradient, your data points should span a wide portion of the graph. If a taller stand is available, starting at $h = 0.750\,\text{m}$ or higher is even better, because it extends the range and reduces the percentage uncertainty in height measurements.

Part 4: Expected Results, Graphs, and Interpretation

Sample results table

The table below shows realistic data for this experiment using the two-light-gate method. These are adapted from handbook sample results for the practical.

Notice that as $h$ decreases, $t$ decreases (shorter fall, less time) and $2h/t$ also decreases because the average velocity over a shorter drop is lower.

The graph: $2h/t$ against $t$

The graph below shows the straight-line pattern to expect; notice that the gradient gives $g$ while the positive $y$-intercept represents $2u$ rather than zero.

[DIAGRAM: asset_name: RP 03 - Determination of g by a Free-Fall Method - Diagram 2; asset_slug: RP 03 - Determination of g by a Free-Fall Method - Diagram 2; recommended_method: retained_png; description: Graph with $t$ / s on the x-axis (range 0.14 to 0.24) and $2h/t$ / m s$^{-1}$ on the y-axis (range 3.3 to 4.4). Six data points forming a straight line with a positive gradient. A best-fit line drawn through the points, extended to show the y-intercept at approximately $2u \approx 2.0\,\text{m s}^{-1}$ (so $u \approx 1.0\,\text{m s}^{-1}$). The gradient is labelled as $g$. Axes labelled with units. Title: "Graph of $2h/t$ against $t$ for determination of $g$".]

The graph should produce a straight line because the equation $\frac{2h}{t} = gt + 2u$ is linear in $t$. Key features to note:

• Gradient = $g$ (the acceleration due to gravity). You should obtain a value close to $9.81\,\text{m s}^{-2}$.
• $y$-intercept = $2u$, the initial velocity of the ball at the upper light gate multiplied by 2. This should be a positive value set by the fixed gap between the electromagnet and the upper light gate.
• The line should not pass through the origin because $2u \neq 0$.

The physical interpretation of the positive gradient is straightforward: as the fall distance increases, the ball has more time to accelerate, so both $t$ and the average speed increase. The ratio $2h/t$ increases linearly with $t$ because the acceleration is constant.

Alternative graph: $h$ against $t^{2}$ (for the trapdoor method)

If $u = 0$ (trapdoor setup), plot $h$ on the $y$-axis against $t^{2}$ on the $x$-axis. This gives a straight line through the origin with gradient $\tfrac{1}{2}g$, so $g = 2 \times \text{gradient}$.

The handbook sample results for this method are:

These values are systematically high (around $10.7\,\text{m s}^{-2}$ on average versus $9.81\,\text{m s}^{-2}$), so this particular dataset is not showing a release delay. A release delay from residual magnetism would make the measured times too long and would push the calculated value of $g$ lower, not higher. High values of $g$ instead suggest some combination of slightly underestimated times, slightly overestimated heights, or a timing interval that did not begin from true rest.

Part 5: Worked Example — Full Calculation

Here is the complete analysis using the sample data from Part 4 (two-light-gate method).

Step 1: Calculate processed quantities

The quantity $2h/t$ has already been computed in the results table. For example, at $h = 0.500\,\text{m}$ and $t = 0.233\,\text{s}$:

Step 2: Determine the gradient

Using two well-separated points on the best-fit line (not just data points, but points read from the line itself):

• Point 1 (from the line): $t_{1} = 0.150\,\text{s}$, $(2h/t)_{1} = 3.47\,\text{m s}^{-1}$
• Point 2 (from the line): $t_{2} = 0.230\,\text{s}$, $(2h/t)_{2} = 4.25\,\text{m s}^{-1}$

When reading the gradient from a graph, use two points that are far apart on the best-fit line rather than two original data points. A large gradient triangle reduces percentage reading error.

Step 3: State the result

Since gradient $= g$:

This is close to the accepted value of $9.81\,\text{m s}^{-2}$, which is what you would hope for from a well-aligned light-gate setup using a dense ball bearing and a sensible range of heights.

Step 4: Extract the $y$-intercept

Extrapolating the best-fit line to $t = 0$:

Using the line equation: $2h/t = 9.75t + c$. Substituting one point:

So $2u = 2.01\,\text{m s}^{-1}$, giving $u = 1.01\,\text{m s}^{-1}$. This is a plausible velocity for a ball that has already fallen a short distance before reaching the upper light gate.

Step 5: Percentage difference

A well-conducted experiment with good light gates and a tall stand should achieve a value within a few per cent of the accepted result. If your graph produced something much lower, that would point to significant error in the data or in the line of best fit rather than to the expected outcome of the method.

Part 6: Uncertainty and Error Analysis

Uncertainty analysis tells you whether your value of $g$ is convincing. You need to identify, classify, quantify, and improve the important sources of error.

Systematic errors in this practical

Residual magnetism in the electromagnet: When the current is switched off, the core can retain magnetism for a short time. In the two-light-gate method, that delay happens before the timer starts, so by itself it does not change the measured fall time $t$. In Method B, however, the timer may already be running while the ball is still attached, making every recorded time too long and the calculated value of $g$ too low.

Electronic switching delay: There can be a small delay in the release circuit or impact switch. This matters far more in Method B than in Method A, because Method A only times the motion between the two light gates.

Release not perfectly clean: If the ball does not detach from the electromagnet in exactly the same way every time, the initial speed $u$ at the upper light gate can vary slightly. In the two-light-gate method this mainly changes the intercept rather than the gradient, provided the gap to the upper gate is kept constant.

Random errors in this practical

Variation in ball release: The ball may not detach from the electromagnet identically every time, leading to slightly different initial velocities. This causes scatter in $t$ values at each height.

Parallax in reading the ruler: When measuring $h$, if the observer's eye is not level with the light gate beam, the apparent position on the ruler shifts. This introduces random scatter in $h$.

Electronic noise in the timer: The light gate and timer system may trigger slightly differently depending on exactly how the ball passes through the beam (e.g., if it spins or wobbles).

Worked uncertainty calculation

Let us work through the percentage uncertainty in $g$ from this experiment.

Uncertainty in $h$:

The metre ruler has 1 mm graduations. You read two positions (upper and lower gate) each with an uncertainty of $\pm 0.5\,\text{mm}$. The total uncertainty in $h$ is:

For the smallest height $h = 0.250\,\text{m}$:

Uncertainty in $t$:

The electronic timer has a resolution of $\pm 1\,\text{ms} = \pm 0.001\,\text{s}$. For the shortest time $t = 0.146\,\text{s}$:

However, the dominant uncertainty in $t$ is more likely the random variation between repeats. If three readings at a given height are $0.144$, $0.146$, and $0.148\,\text{s}$, the range is $0.004\,\text{s}$ and the uncertainty is:

Percentage uncertainty in $g$:

Since $g$ is determined from the gradient of the $2h/t$ vs $t$ graph, the uncertainty in $g$ is best found from the worst acceptable line (also called the steepest or shallowest line that still passes through the error bars of the data points).

If the best-fit gradient is $9.6\,\text{m s}^{-2}$ and the worst acceptable line gives $10.4\,\text{m s}^{-2}$:

A well-conducted experiment should achieve a percentage uncertainty in $g$ of around 5-10%. If your uncertainty is larger, this suggests excessive scatter in your data.

To estimate the uncertainty in the gradient, draw a worst acceptable line: the steepest or shallowest straight line that still passes through the error bars. Compare its gradient with the best-fit gradient to judge the likely uncertainty in $g$.

Sources of error and improvements

In 4-6 mark evaluation questions, the best answers follow a clear chain: name a specific error, classify it if asked, say whether it makes $h$ or $t$ too large or too small, then explain how that changes the calculated value of $g$. A vague phrase such as "human error" is not enough.

Part 7: Judging the Result

Once you have a value for $g$, the next job is to decide whether the result is convincing. A strong conclusion does more than quote a number. It checks whether the graph is straight, whether the intercept is sensible, and whether the uncertainty is small enough for the accepted value to lie within the likely range of results.

A careful write-up should comment on four things:

• the plotted points lie close to a straight line, supporting the model $\dfrac{2h}{t} = gt + 2u$
• the gradient gives a value of $g$ reasonably close to $9.81\,\text{m s}^{-2}$
• the intercept is positive and of a sensible size, showing that the ball was already moving at the upper light gate
• any remaining discrepancy is explained using specific sources of error rather than vague phrases such as "human error"

Common question-family traps

• In Method A, $h$ is the separation between the two light gates, not the full distance from the electromagnet to the lower gate.
• In Method A, the useful straight-line plot is $2h/t$ against $t$, not $h$ against $t^{2}$.
• In Method A, a positive intercept is expected because it equals $2u$; the line should not be forced through the origin.
• Repeats reduce random error, but they do not remove a systematic timing offset.
• An improvement only scores well if you link it to a named error and explain how it reduces that error.

This practical also links naturally to the rest of mechanics. The same constant acceleration $g$ appears again in free fall and projectile motion, and the same graph skills are used whenever you linearise data to extract a physical constant.