1.3-1.5 - Distance-time graphs, speed and motion practical

What distance-time graphs show

A distance-time graph shows how far an object is from a starting point, or how much distance it has moved, at different times. Always read the axis label carefully, because an exam may use distance from start, distance travelled, or simply $distance$.

On a standard distance-time graph:

• time is on the horizontal axis
• distance is on the vertical axis
• each plotted point matches one time reading with one distance reading

To plot one, choose sensible scales, plot points accurately with small crosses, and draw a straight line or smooth curve that represents the pattern. Do not join points with a jagged line unless the data genuinely shows abrupt changes. If a point does not fit the pattern, repeat the measurement and investigate the cause before deciding whether to exclude it.

The graph is not just a picture. It is a compact record of motion: the vertical change tells you how much the distance changed, and the horizontal change tells you how much time passed.

Explaining graph shape

The shape of a distance-time graph tells you how the motion changes.

A horizontal line means the distance is not changing. The object is stationary for that time interval.

A straight sloping line means the distance changes by equal amounts in equal times. The object is moving at constant speed.

A steeper line means a greater distance is covered in the same time. The speed is larger.

A curved line means the speed is changing. If the curve gets steeper, the object is speeding up. If it gets less steep, the object is slowing down. For this lesson, connect changing steepness to changing speed.

If the vertical axis is distance from start, a downward sloping line means the object is moving back towards the starting point. If the vertical axis is total distance travelled, the graph cannot go down, because total distance travelled only stays the same or increases.

Good graph explanations use interval language. Instead of saying "the graph goes up", say "from 0 s to 4 s the object moves away from the start at constant speed."

Average speed

Average speed tells you the distance moved per second, minute or hour over a whole journey or time interval. It does not say that the object moved at exactly that speed at every instant.

If distance is in metres and time is in seconds, the speed is in metres per second, written $m/s$ or $m\,s^{-1}$. If distance is in kilometres and time is in hours, the speed is in kilometres per hour, written $km/h$.

For example, a tennis ball moves $1.8\text{ m}$ in $0.60\text{ s}$.

The formula can be rearranged if a different quantity is needed:

The distance in the formula is the total distance moved during the interval. If an object moves away from a start point and then returns, its final distance from the start is not the same as the total distance it moved. In the same way, the time taken is the whole time for the interval, including any time the object is stationary.

Gradient is speed

For a straight section of a total-distance-travelled graph, speed is the gradient. For straight-line motion shown as distance from the start, use the magnitude of the gradient: a negative gradient indicates return towards the start, not a negative speed.

[DIAGRAM: asset_name: Distance-time graphs, speed and motion practical - diagram 01; asset_slug: p02_distance_time_graphs_speed_and_motion_practical__diagram_01; recommended_method: matplotlib; description: White 16:9 NovaLearn pale-palette diagram with grey outlines and dark-grey labels. Blue distance-time journey through (0,0), (4,8), (7,8), (10,20), (13,8), with an unobscured clay gradient triangle and adjacent section labels.]

To calculate a gradient:

1. Choose two clear points on the straight section, as far apart as possible: a large gradient triangle gives a more accurate answer.
2. Find the change in distance between them.
3. Find the change in time between them.
4. Divide change in distance by change in time.

For the first sloping section in the diagram, the distance changes from $0\text{ m}$ to $8\text{ m}$ while the time changes from $0\text{ s}$ to $4\text{ s}$.

The same method works for every straight section. From $4\,\mathrm{s}$ to $7\,\mathrm{s}$ the line is horizontal, so the gradient and the speed are zero. From $7\,\mathrm{s}$ to $10\,\mathrm{s}$ the distance changes from $8\,\mathrm{m}$ to $20\,\mathrm{m}$, so the speed is $(20-8)/(10-7)=4.0\,\mathrm{m/s}$: the steeper line shows the greater speed. On the returning section shown, the gradient is $(8-20)/(13-10)=-4.0\,\mathrm{m/s}$. The speed is $4.0\,\mathrm{m/s}$; the minus sign describes motion towards the start. The full journey covers $8+12+12=32\,\mathrm{m}$, so its average speed is $32/13=2.46\,\mathrm{m/s}$, not $8/13$: the final distance from the start is not the distance travelled.

For a curved graph, the speed is not constant. To estimate speed at one instant, draw a tangent touching the curve at that point. Choose two widely separated points on the tangent, read their coordinate changes and calculate the magnitude of $\Delta d/\Delta t$. Use points on the tangent, not two arbitrary points on the curve: a chord gives an average over an interval. You can still compare speeds qualitatively by comparing steepness: the steeper part of the curve shows the greater speed.

Investigating motion

The required practical is to investigate the motion of everyday objects, such as toy cars or tennis balls. A simple method is to measure the time taken for an object to travel known distances, then use the data to calculate average speeds and plot a distance-time graph.

For a toy car:

1. Set up a ramp or track and mark several distances along it, such as $0.20\text{ m}$, $0.40\text{ m}$, $0.60\text{ m}$, $0.80\text{ m}$ and $1.00\text{ m}$.
2. Release the same toy car from the same starting position each time, without pushing it.
3. Start timing when the front of the car passes the start mark.
4. Stop timing when the front of the car reaches the chosen distance mark.
5. Repeat each distance at least three times and calculate a mean time, leaving out any anomalous repeat that is far from the others.
6. Plot distance on the vertical axis against mean time on the horizontal axis.

For a tennis ball drop, measure the fall distance and use slow-motion video with a known frame rate: count frames from release to the ball reaching the finish mark and divide by frames per second. A short drop is too quick for reliable hand timing. Keep the camera perpendicular to the ruler, use the same point on the ball, repeat drops, and keep the landing area clear. For a car, two light gates can time passage between known positions; timing one interrupt card through one gate measures speed at that gate instead of the whole journey.

The independent variable is the quantity you deliberately change, such as the distance mark or ramp height. The dependent variable is the quantity you measure, such as time taken or average speed. Control variables are kept the same, such as the toy car, release point, ramp surface and release method.

If the graph is a straight line through the data, the motion has constant speed over that range. If the graph curves, the speed is changing. If a car rolls down a ramp, it often speeds up, so one calculation of distance divided by time gives only the average speed for that whole run.

Improving and using data

Practical answers in Physics need more than "repeat it". You should be able to say what the main measurement problem is and how your method reduces it.

With stopwatch timing, human reaction time is often a large uncertainty. A typical reaction time is about $0.2\,\mathrm{s}$, and it may not be the same when you start and stop the stopwatch. This matters most when the time interval is short: $0.2\,\mathrm{s}$ is a large fraction of a $0.86\,\mathrm{s}$ run, but a small fraction of a $5\,\mathrm{s}$ run. A longer measured interval reduces the fractional effect of a similar timing uncertainty. Repeating and averaging reduces random variation but does not remove a consistent timing bias. Light gates or video timing avoid hand reaction time; a clear start/finish marker makes the event being timed consistent.

Distance measurements also need a consistent reference point. For a toy car, measure to the front of the car each time. For a ball, measure from the same part of the ball each time. Mixing reference points changes the measured distance and makes the results less valid.

A useful results table includes the distance, repeat times, mean time and calculated average speed:

The increasing average speed in this table suggests the car is not moving at constant speed over the measured runs. The data is more convincing because the repeat times are close together and the trend is consistent.

When evaluating a conclusion, link it to the evidence. "The car went faster" is weak. "The calculated average speed increased from $0.47\text{ m/s}$ to $0.71\text{ m/s}$, so the car moved faster over the longer runs" is stronger because it quotes data.

Quick recap

• Average speed is total distance travelled divided by total time; final distance from the start is not necessarily total distance travelled.
• A horizontal distance-time section means stationary; a straight slope means constant speed. Use gradient magnitude for speed on a distance-from-start graph.
• A tangent estimates speed at one instant on a curve; a gradient over an interval describes average motion over that interval.
• Measure consistent positions and times, control the release conditions, and repeat readings. Video or light gates avoid hand reaction-time errors.