Motion apparatus and data

An investigation of motion needs a way to measure position or distance and a way to measure time. Different apparatus gives different kinds of data, so the best choice depends on the motion being studied.

For motion and collision experiments, light gates or video can measure speeds before and after an interaction. A card of known length passing through a light gate gives:

The symbol $\approx$ is important. The value is an average speed while the beam is blocked, not an exact instantaneous speed. The shorter the card or diameter, and the smaller the timing uncertainty, the closer the value is to the speed at that position.

When planning, name the variables clearly. The independent variable is the quantity deliberately changed, the dependent variable is the quantity measured as the outcome, and control variables are kept the same to make the test valid. For example, in a trolley experiment down a runway, the slope could be the independent variable, acceleration the dependent variable, and trolley mass/runway surface/release method control variables.

Free fall and g

Free fall has a precise meaning: the only significant force on the object is its weight. Near the Earth's surface, an object in free fall has a downward acceleration with magnitude about $9.81\ \text{m s}^{-2}$.

In a school laboratory, free fall is an ideal model. A dense steel ball dropped through a short distance is a reasonable approximation because air resistance is small compared with its weight. An inflated balloon is not a good approximation: air resistance is large compared with its weight, so the measured acceleration would be much smaller than $g$.

The acceleration $g$ is independent of the falling object's mass when air resistance is negligible. A heavier object has a larger weight, but it also has more mass to accelerate. These effects balance in the ideal model, so mass alone does not make one object have a larger free-fall acceleration.

For a dropped object released from rest, the applied analysis often uses:

Here $h$ is the vertical distance fallen in metres and $t$ is the time in seconds. This is the constant-acceleration equation $s = ut + \frac{1}{2}at^2$ with $u = 0$, $s = h$, and $a = g$, using downward as positive.

Electromagnet and trapdoor method

One method for determining $g$ releases a steel ball using an electromagnet. The same switching action starts the timer. When the ball reaches the trapdoor switch below, the timer stops.

A method that can produce useful data is:

1. Hold a steel ball at the electromagnet and place the trapdoor directly below.
2. Measure the vertical distance $h$ from the release point to the point that triggers the stop signal. Use a plumb line or set square to keep this distance vertical.
3. Release the ball without pushing it, so the initial velocity is as close to zero as possible.
4. Record the time $t$ for the fall.
5. Repeat at the same height to identify anomalies and calculate a mean time.
6. Repeat for several different heights and process the data using a graph.

The graph route is stronger than relying on one drop. From:

a graph of $h$ on the vertical axis against $t^2$ on the horizontal axis should be a straight line through the origin. Its gradient is:

so:

Worked example: using a single drop as a first estimate

A steel ball is released from rest and falls through $1.20\ \text{m}$. Three measured times are $0.497\ \text{s}$, $0.492\ \text{s}$ and $0.495\ \text{s}$.

Mean time:

For a first estimate:

so:

Substitute:

This agrees very closely with the accepted value, but a graph from several heights would be more reliable because it uses more data and reduces the effect of one poorly timed drop.

Light gates and timers

Light gates can be used in two related ways. A single gate can find the speed of a falling object as it passes the gate. Two gates can find speeds at two positions and the time between those positions.

For one light gate:

where $d$ is the diameter of the ball or the length of the card blocking the beam, and $\Delta t$ is the beam-block time. If a ball is falling, the speed is increasing during the block interval, so this is an average speed near the gate.

For two light gates:

1. Measure the separation $s$ between the gates using a consistent vertical reference point.
2. Use the gate times to determine the speed $u$ at the first gate and $v$ at the second gate.
3. Use the timer or data logger to measure the time interval $t$ between the two speed measurements.
4. Determine the acceleration using:

If air resistance is negligible and the object was falling freely between the gates, this acceleration is an experimental value for $g$.

Worked example: two light gates

A ball of diameter $16.0\ \text{mm}$ falls through two light gates. At gate 1, the beam-block time is $7.60\ \text{ms}$. At gate 2, the beam-block time is $5.00\ \text{ms}$. The time between the two speed measurements is $0.110\ \text{s}$.

Convert the diameter and times:

Speed at gate 1:

Speed at gate 2:

Acceleration between the gates:

This is close to $9.81\ \text{m s}^{-2}$, so the data are consistent with free fall within normal laboratory uncertainty.

Uncertainty, errors and improvements

Determining $g$ is not complete when a number appears on the calculator. A good practical answer also discusses how trustworthy the number is.

Important sources of uncertainty and error include:

• distance measurement: parallax, measuring from inconsistent points, or a drop path that is not vertical
• timing: human response time for stopwatch methods, trigger delay, or finite beam-block length in light-gate methods
• release method: pushing the object accidentally gives a non-zero initial velocity
• air resistance: drag reduces the measured acceleration, especially for light or large-area objects
• alignment: the object may miss the gate centre, clip the trapdoor unevenly, or move sideways

Useful improvements follow directly from these limitations. Use a dense small object, release it without a push, align the drop vertically with a plumb line, use electronic timing, repeat readings, use several heights, plot a best-fit line, and use a large gradient triangle rather than two nearby plotted points.

Safety is simple but still real: secure clamps and stands, use a catcher or tray for falling balls, keep feet and hands away from the impact region, and avoid placing fragile sensors where the object can strike them.

When comparing an experimental value with an accepted value, use percentage difference:

Small percentage difference suggests good accuracy, but it does not prove the method is valid by itself. Two systematic errors can sometimes cancel. Always connect the numerical comparison to the method.