A solve-object-distance rearrangement scan holds the image distance fixed and varies focal length across rows, inferring the object distance from the rearranged lens equation u = fv / (v - f). The arithmetic is exact only for the stated object and image distances, focal lengths, magnifications, signed heights, and rows in the ideal thin-lens model. Real measurements add alignment, lens aberrations, finite aperture, thickness, screen/readout calibration, sign-convention bookkeeping, finite precision, and measurement uncertainty.
Rearrange the lens equation to infer object distance from a known image distance and focal length. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
Use image distance to infer object distance
Sometimes the image screen position is known first. Keep the image distance at 30 metres and ask what object distance would make each focal length land there.
f1=u1+v1
Rearrange for the object distance
Move the image-distance reciprocal across the equation. The object distance is set by the remaining reciprocal budget.
u1=f1−v1⇒u=v−ffv
Three focal lengths need three object positions
Hold the image distance and object height fixed at 4 metres. A longer focal length leaves less reciprocal room for the object, so the object must sit farther from the lens.
For the middle row, the object and image distances both sit at 30 metres. The magnification is -1, so a positive object height becomes an equal-size inverted image.