Coaching Centre
Investigating projectile motion
The mandatory projectile-motion practical: rolling a ball off a bench, measuring with carbon paper, video or strobe photography, linearising data to find g, and evaluating validity, reliability and accuracy.
Around 1608, Galileo rolled bronze balls down a ramp and off the edge of a table, then marked where they landed. His working page, now in a library in Florence, shows the landing distances lining up with a half-parabola. It was one of the first experiments in history to test a mathematical model of motion, and it is basically the practical you do in Year 12.
The syllabus requires you to collect your own data that validates the projectile relationships from 5A and 5B. Exam questions on this practical usually ask you to describe the method, process the results, or evaluate validity, reliability and accuracy. This page covers all three.
A typical method
Processing the results
From the landing points, plot measured range against launch speed. The model predicts a straight line through the origin with gradient $\sqrt{2h/g}$. For an 80.0 cm bench, that gradient is $\sqrt{1.6/9.8} = 0.404$ s, which is the time of flight.
From a video or strobe photo, you get position at equal time intervals. Three graphs each test one part of the model:
| Plot | Model predicts | Gradient gives |
|---|---|---|
| $x$ against $t$ | straight line (constant $u_x$) | $u_x$ |
| $v_y$ against $t$ | straight line (constant acceleration) | $g$ |
| $y$ against $t^2$ (dropped distance) | straight line through origin | $g/2$ |
The last one is called linearising. A graph of $y$ against $t$ is a curve, and it's hard to judge whether a curve is exactly a parabola. Since $y = \tfrac12 g t^2$, plotting $y$ against $t^2$ should give a straight line, and a straight line is easy to judge and to fit.
A student's graph of vertical drop $y$ (m) against $t^2$ (s²) has a line of best fit with gradient 4.85 m s⁻². Find $g$ and the percentage difference from 9.80 m s⁻².
- Choose the middle release (1.2 m/s) and take a strobe photo. Before you measure, check the predicted landing distance: the ball should land 48.5 cm from the bench.
- Press Measure positions. On the "x against t" graph, how close is the gradient to 1.2 m/s?
- Switch to "vertical velocity against t" and then "y against t²". Which method gives a value of g closer to 9.8 m/s²? Take a few new photos (each has new measurement errors) to see which is more consistent.
- Change the release point. Does g change? Should it?
Validity, reliability and accuracy
These three words have specific meanings in HSC marking, and mixing them up costs marks.
- Validity: does the experiment actually test the relationship? It needs the ball to leave the bench horizontally, the launch speed to be the same each trial from a given point, and air resistance to be negligible (a small dense ball, short flight). Controlled variables: bench height, the ball, the release point.
- Reliability: do repeated trials give consistent results? Repeating each release five times and checking the spread of landing marks shows this. A tight cluster of marks means reliable data.
- Accuracy: how close is the result to the accepted value? Compare your measured $g$ or range with the prediction, as a percentage difference.
Common errors and fixes:
| Problem | Type | Improvement |
|---|---|---|
| Launch speed measured on the bench, but the ball slows slightly before the edge | systematic | Measure the speed right at the edge with a light gate |
| Range measured from the bench leg, not directly below the edge | systematic | Use a plumb line to mark the launch point on the floor |
| Ball released by hand at slightly different points | random | Use a stop block on the ramp; repeat and average |
| Parallax when reading a video against a ruler | random or systematic | Place the grid in the same plane as the ball; camera far away and level |
| Ramp not horizontal at the edge | systematic | Check the bottom of the ramp with a spirit level |
Repeating trials improves reliability and reduces the effect of random errors. It does nothing for a systematic error, which shifts every result the same way. You fix systematic errors by changing the method or equipment.
Test yourself
A student plots vertical drop $y$ against $t^2$ for a falling ball and gets a straight line with gradient 4.9 m s⁻². The value of $g$ is
$y = \frac12 g t^2$, so $g = 2 \times$ gradient.
Which change would improve the reliability of the investigation?
Reliability is about consistency of repeated measurements.
A ball leaves an 80 cm bench horizontally at 1.5 m/s. Its predicted range is closest to
$t = \sqrt{2(0.80)/9.8} = 0.404$ s, so $\Delta x = 1.5 \times 0.404 = 0.606$ m.
Every measured range is about 2 cm shorter than predicted. The most likely cause is
An error that shifts every result the same way is systematic.
Label the equipment for the projectile practical.
Which does each change mainly improve?
Plotting $y$ against $t$ gives a curve, so we the data by plotting $y$ against . The gradient of this graph is half of . Repeating trials reduces the effect of random errors but not errors. A graph of $x$ against $t$ is a straight line because the horizontal is constant.
A student rolls a ball down a ramp held by hand at the edge of a bench. The student measures the launch speed once, using a stopwatch over 1.0 m of bench, then measures the landing distance once with a tape measure from the bench leg. The measured range is 8% less than predicted. Evaluate the method and suggest improvements.
The method has low validity, reliability and accuracy, so the 8% difference does not show the model is wrong.
Holding the ramp by hand means the ball may not leave the bench horizontally and the release changes from trial to trial. Clamping the ramp, checking the bottom is level, and using a stop block would fix both. Timing 1.0 m with a stopwatch adds reaction-time error of around 0.2 s, and the ball slows on the bench, so the speed at the edge is lower than measured, which explains a range shorter than predicted (a systematic error). A light gate at the edge, or video against a scale, would measure the true launch speed. Measuring from the bench leg rather than directly below the edge adds another systematic error; a plumb line fixes it. With only one trial there is no way to judge reliability; five repeats from the same release point, landing on carbon paper, would show the spread and allow a mean. With these changes the comparison with the prediction would be a valid test.