Measuring One-Dimensional Motion for HSC Physics

A trolley rolls down a track. You record its position every 0.20 s, but the numbers jump around slightly because your measurements are not perfect. How do you turn that messy table into a sensible statement about its motion? More importantly, how do you tell whether it is moving at constant velocity, accelerating, or simply suffering from noisy data?

That is the real job of measuring motion. The experiment does not hand you “velocity” or “acceleration”. It gives you observations. You have to collect those observations carefully, organise them, and extract the physics.

01What do we actually measure?

Imagine filming a toy car moving from left to right along a straight track.

Pause the video at several moments. At each frame, you can record two things directly:

• the time, \(t\), usually in seconds (s)
• the position, \(x\), usually in metres (m)

A simple table might look like this:

Before reading further, predict the motion.

Is the car speeding up, slowing down, or moving at constant velocity?

The position increases by 0.20 m every 0.50 s. The same displacement occurs in the same amount of time, so the velocity is constant.

The important point is that position is normally the easiest motion quantity to measure directly. Velocity and acceleration are often calculated from how position changes.

If position, displacement, and direction still feel slippery, review distance and displacement before continuing.

02Start by choosing a coordinate system

Before collecting data, decide what your position numbers mean.

Suppose a trolley can move in either direction along a track. You might choose:

• the starting point as \(x = 0\)
• movement to the right as positive
• movement to the left as negative

Then:

• \(x = +2.0\text{ m}\) means the trolley is 2.0 m to the right of the origin
• \(x = -0.5\text{ m}\) means it is 0.5 m to the left

This choice is arbitrary. Physics does not care whether you call right positive or left positive.

It does care that you stay consistent.

Check your understanding

A student chooses left as the positive direction. A trolley moves 1.2 m to the right.

What is its displacement?

03Ways of collecting one-dimensional motion data

Different practical setups give you different kinds of data.

Stopwatch and ruler

The simplest method is to mark known positions along a track and record when an object reaches each mark.

For example:

This works, but human reaction time can be a serious source of uncertainty.

If the entire motion lasts only 0.5 s, a timing error of 0.1 s is enormous. If the motion lasts 20 s, the same timing error matters much less.

So “use a stopwatch” is not automatically a good experimental design. The useful question is: is the timing method precise enough for the timescale of the motion?

Video analysis

Video is often much better for short motions.

You place a ruler or another object of known length in the same plane as the motion, record the object, then measure its position frame by frame.

If the video is recorded at 50 frames per second, the time between consecutive frames is:

\[
\Delta t = \frac{1}{50} = 0.020\text{ s}
\]

Video analysis gives many position-time measurements from a single run.

But there is a trap.

Imagine filming a railway track from an angle. Objects farther from the camera appear smaller. A distance of 10 cm in the image no longer represents the same real distance everywhere.

To reduce this problem:

• keep the camera fixed
• point it as squarely as possible at the direction of motion
• place the scale in the same plane as the moving object
• track the same point on the object in every frame

The last point sounds minor, but it matters. If you track the front of the trolley in one frame and its centre in the next, you have created fake motion.

Light gates

A light gate measures the time at which an object interrupts a light beam.

With two light gates a known distance apart, you can determine an average velocity between them.

If an interrupt card of known length passes through one light gate, you can estimate the trolley’s velocity at that location:

\[
v \approx \frac{L}{\Delta t}
\]

where:

• \(v\) is the velocity in metres per second
• \(L\) is the length of the card in metres
• \(\Delta t\) is the time for the card to pass through the gate in seconds

This is only an approximation to instantaneous velocity because the trolley moves a small distance while the card passes through the beam. A shorter card measures the motion over a smaller interval.

Motion sensors

A motion sensor can record position automatically at many times per second.

That sounds ideal, but more data does not automatically mean better data.

A sensor might produce readings such as:

\(1.102\text{ m}\), \(1.099\text{ m}\), \(1.105\text{ m}\), \(1.101\text{ m}\)

even when an object is almost stationary.

Those tiny changes may be measurement noise rather than real motion. This becomes especially important when calculating velocity and acceleration, because both depend on differences between measurements.

04Turning position data into velocity

Suppose an object moves from position \(x_1\) at time \(t_1\) to position \(x_2\) at time \(t_2\).

Its average velocity is:

\[
v_{\text{av}} = \frac{\Delta x}{\Delta t}
= \frac{x_2-x_1}{t_2-t_1}
\]

where:

• \(v_{\text{av}}\) is average velocity in metres per second
• \(\Delta x\) is displacement in metres
• \(\Delta t\) is the time interval in seconds

This tells you how quickly position changed over that interval.

For a deeper distinction between average and instantaneous values, see average and instantaneous velocity.

A tempting mistake: dividing position by time

Students sometimes calculate:

\[
v = \frac{x}{t}
\]

using one position and one time.

That only works in special circumstances, such as when the object started from \(x=0\) at \(t=0\).

The safer relationship is always:

\[
v_{\text{av}} = \frac{x_2-x_1}{t_2-t_1}
\]

Velocity depends on a change in position over a change in time.

05Reading velocity from a position-time graph

Plot position on the vertical axis and time on the horizontal axis.

If the graph is a straight line, the velocity is its gradient:

\[
v = \frac{\Delta x}{\Delta t}
\]

A steeper position-time graph means a larger magnitude of velocity.

A horizontal line has:

\[
\Delta x = 0
\]

so its velocity is zero.

A line sloping downwards has a negative gradient, so the object has negative velocity.

Predict before reading the answer

Two objects have straight position-time graphs. Object A has a steeper upward slope than object B.

Which has the greater velocity?

06What if the position-time graph is curved?

Now suppose the graph becomes steeper as time passes.

The object’s velocity is changing.

A student might try to calculate:

\[
\frac{\text{final position} – \text{initial position}}
{\text{final time} – \text{initial time}}
\]

and call that “the velocity”.

It is a velocity, but specifically the average velocity across the whole interval.

If you want the velocity at one particular moment, you need the gradient of the tangent to the position-time curve at that moment.

With experimental data, however, you usually have separate points rather than a perfectly smooth curve. One useful estimate is to use points on either side of the time you care about.

For equally or nearly equally spaced data:

\[
v_i \approx
\frac{x_{i+1}-x_{i-1}}
{t_{i+1}-t_{i-1}}
\]

Here \(v_i\) is the estimated velocity at the middle time.

This is called a central difference estimate.

You do not need the name to understand the idea. You are measuring how much position changes across a short interval centred on the time of interest.

07Turning velocity data into acceleration

Acceleration tells you how quickly velocity changes.

Average acceleration is:

\[
a_{\text{av}} = \frac{\Delta v}{\Delta t}
= \frac{v_2-v_1}{t_2-t_1}
\]

where:

• \(a_{\text{av}}\) is acceleration in metres per second squared, \(\text{m s}^{-2}\)
• \(\Delta v\) is the change in velocity in \(\text{m s}^{-1}\)
• \(\Delta t\) is the time interval in seconds

The unit \(\text{m s}^{-2}\) can feel strange at first.

An acceleration of \(2.0\text{ m s}^{-2}\) means the velocity changes by \(2.0\text{ m s}^{-1}\) every second.

For example:

• after 1 s: velocity has changed by \(2.0\text{ m s}^{-1}\)
• after 2 s: velocity has changed by \(4.0\text{ m s}^{-1}\)
• after 3 s: velocity has changed by \(6.0\text{ m s}^{-1}\)

That is the useful meaning of acceleration.

Check your understanding

A car’s velocity changes from \(3.0\text{ m s}^{-1}\) to \(9.0\text{ m s}^{-1}\) in 2.0 s. Find its average acceleration.

08Acceleration from a velocity-time graph

On a velocity-time graph:

• velocity is on the vertical axis
• time is on the horizontal axis

The gradient is acceleration:

\[
a = \frac{\Delta v}{\Delta t}
\]

A horizontal velocity-time graph has zero gradient, so acceleration is zero.

That does not mean the object is stationary.

If the horizontal line is at \(v = 5\text{ m s}^{-1}\), the object is moving at a constant velocity of \(5\text{ m s}^{-1}\).

This is one of the most common graph-reading mistakes in kinematics.

Misconception: “zero acceleration means zero velocity”

Suppose a cyclist travels steadily at \(8\text{ m s}^{-1}\).

Its velocity is not zero.

But the velocity is not changing, so:

\[
\Delta v = 0
\]

and therefore:

\[
a = 0
\]

Zero acceleration means constant velocity, not necessarily zero velocity.

10Why acceleration data often look messier

Suppose the true positions are:

\(0.500\text{ m}\), \(0.600\text{ m}\), \(0.700\text{ m}\)

but your sensor records:

\(0.501\text{ m}\), \(0.598\text{ m}\), \(0.703\text{ m}\)

Those errors are only a few millimetres.

That seems harmless until you subtract nearby measurements to calculate velocity.

Then you subtract the velocity measurements again to calculate acceleration.

Small position errors can therefore become much more noticeable in acceleration data.

Think of it like trying to judge whether your friend’s walking pace changed by watching their location. Estimating where they are is fairly easy. Estimating their speed is harder. Estimating how quickly their speed is changing is harder again. Each step depends on differences between already measured quantities.

The analogy breaks because experimental calculations are numerical rather than visual guesses, but the same basic problem remains: acceleration is more sensitive to measurement noise.

This is why an acceleration graph from real data often looks less tidy than the textbook version.

11Use the graph that matches the question

Choosing the right graph is part of analysing motion.

Do not just calculate everything because the calculator can.

Ask what the experiment is trying to determine.

If you expect constant velocity, plotting position against time and fitting a straight line is often much better than calculating ten separate velocities.

The gradient of the best-fit line gives an estimate using the whole data set rather than just two points.

12Best-fit lines are usually better than joining the dots

Real measurements scatter.

Suppose an object should move at constant velocity, but your measured positions are slightly above or below the ideal straight line.

If you join every point, you may create a jagged graph that suggests the object’s velocity changed repeatedly.

That may be an artefact of measurement.

If the physical model predicts a linear relationship, draw or calculate a line of best fit instead.

Its gradient gives:

\[
v = \frac{\Delta x}{\Delta t}
\]

using the overall trend.

When finding the gradient of a best-fit line, choose two points that:

• lie on the line, not necessarily on measured data points
• are far apart

Using widely separated points reduces the effect of small reading errors.

Check your understanding

A position-time best-fit line passes through \((1.0\text{ s}, 0.8\text{ m})\) and \((5.0\text{ s}, 4.4\text{ m})\). Find the velocity.

13Do not confuse precision with accuracy

Suppose five measurements of a stationary object’s position are:

\(1.243\text{ m}\), \(1.244\text{ m}\), \(1.243\text{ m}\), \(1.244\text{ m}\), \(1.243\text{ m}\)

They are tightly grouped, so they are precise.

But suppose the object is actually at \(1.300\text{ m}\), and the measuring system was incorrectly calibrated.

The measurements are precise but inaccurate.

For a motion practical, useful checks include:

• Is the ruler or scale calibrated correctly?
• Is the timer starting at the correct event?
• Is the camera aligned properly?
• Is the sensor detecting the object you think it is detecting?
• Are measurements being recorded with sensible significant figures?
• Is the time interval small enough to reveal the motion you care about?

Good data collection starts before the object moves.

14Sampling rate changes what you can see

Imagine a ball that speeds up and slows down over 1.0 s.

If you record its position only at:

• \(t=0\text{ s}\)
• \(t=1.0\text{ s}\)

you know almost nothing about what happened between those moments.

The ball could have moved smoothly. It could have stopped halfway. It could have sped up and then slowed down.

A higher sampling rate gives you more information about changes within the motion.

But making the interval extremely small creates another problem. The displacement between consecutive readings may become similar in size to the measurement uncertainty.

Suppose the position uncertainty is a few millimetres, and the object moves only 2 mm between frames. Calculating velocity from those frames will be unreliable.

So there is a trade-off:

• intervals that are too large can hide changes
• intervals that are too small can make measurement noise dominate

The best interval depends on how fast the object moves and how precise your equipment is.

15A practical workflow for analysing one-dimensional motion

When you are given experimental motion data, work through it in this order.

1. Define the axis. Decide the origin and positive direction.
2. Record time and position. Include units in the table headings.
3. Inspect the raw data. Look for impossible values, sudden jumps, or obvious recording errors.
4. Plot position against time.
5. Examine the shape. A straight line suggests constant velocity. A changing gradient suggests changing velocity.
6. Calculate velocity if required. Use gradients or changes in position divided by changes in time.
7. Plot velocity against time if acceleration matters.
8. Calculate acceleration from the velocity-time gradient.
9. Compare the data with the expected model.
10. Discuss uncertainty before blaming every irregular point on new physics.

That last step matters. One strange measurement does not automatically mean the trolley experienced a mysterious force for 0.02 s.

16One final misconception: the graph is not a picture of the path

A position-time graph that slopes upwards is not showing a trolley climbing a hill.

The horizontal axis is time, not horizontal distance.

Likewise, a curved position-time graph does not mean the object travelled along a curved physical path. The motion can still be completely one-dimensional.

The shape of the graph tells you how the measured quantity changes with time.

Once you can move confidently from time and position measurements to velocity and acceleration, you can test much stronger claims about motion. In particular, you can decide from real data whether an object is undergoing uniform motion or uniformly accelerated motion, rather than simply assuming that it is.