Motion Graphs Camera Lab

A camera behaves like a pinhole: the further away an object is, the smaller its image. By similar triangles, the image width is proportional to 1 ÷ distance. Image area is proportional to 1 ÷ distance², so the square root of the area works too.

Here A is the number of pixels the coloured sheet covers. Calibrating at a measured distance d₀ fixes the constant: k = d₀√A₀.

Far from the camera the sheet covers few pixels, so a single pixel of error matters more. That is why readings are noisier beyond about 3 m.

Velocity is the gradient of the displacement–time graph. Acceleration is the gradient of the velocity–time graph.

Each gradient comes from a curve of best fit through the readings either side of that moment. Differentiating amplifies measurement noise, so the a–t graph is the most jagged. More smoothing uses a wider window. It reduces noise but rounds off sudden changes and delays the live graph slightly.

Hover over a recorded graph, or focus it and use the arrow keys. The s–t graph then shows a tangent. The v–t and a–t graphs show the area under the line, which gives the change in displacement or velocity.

• Use a brightly coloured sheet. Bright pink, green or orange works best; white paper is easily confused with walls.
• Remove anything else of that colour from the shot, including clothing.
• Place the camera at chest height with a clear path from about 0.6 m to 3 m.
• Light the room evenly and avoid standing in front of a bright window.
• Hold the sheet flat with both hands, facing the camera, and keep it steady. Walk; do not run.
• Tick “Show what the camera detects”. If the sheet is patchy, raise the tolerance. If the background lights up, lower it.