Projectile Launcher Data That 'Disproves' the Range Equation
In a Grade 11 physics class, three lab groups use identical spring-loaded launchers clamped to the same long bench. Students launch 16 mm steel balls at 35° above horizontal and measure horizontal range with a meter stick taped to the floor. The model predicts R = v0² sin(2θ)/g. Group A measures v0 = 4.9 m/s with a photogate and ranges 2.3, 2.3, 2.4, 2.3 m, matching prediction. Group B, using the same launcher setting, also measures v0 = 4.9 m/s but records ranges 1.9, 1.9, 1.8, 1.9 m. Their protractor reads 35° relative to the bench. Group C's photogate sometimes displays 4.8 m/s and their ranges scatter from 1.6 to 2.5 m. The bench has a visible 1.5° tilt toward the aisle, and the launcher clamp is tight. Group B asks whether their data 'disproves the formula,' and a student suggests deleting the farthest trial as an outlier.
1. Classify the error patterns in Group B and Group C, and evaluate whether the bench tilt can explain Group B's result. Show any calculation you use.
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Group B shows systematic error: four trials cluster within 0.1 m but sit 0.4 m below prediction. Group C shows random error plus an instrument fault: the photogate intermittently reads 4.8 m/s and ranges scatter over 0.9 m.
The bench tilt cannot explain B. Predicted range is R = (4.9² sin70°)/9.8 = 2.30 m, and B is about 17% low. Near 35°, dR/dθ = 2v²cos(2θ)/g ≈ 1.67 m/rad, so 1.5° (0.026 rad) changes range by only ~0.04 m. A 1.5° tilt is therefore a minor correction, not a 0.4 m effect.
Consistent alternatives for B: the ball is not seated to the same depth, the launcher spring is weaker despite the same setting, the protractor angle is measured relative to a tilted bench and the actual launch angle is lower than 35°, or the photogate is overreading speed (e.g., beam crossing a chord rather than the diameter). An actual launch speed of about 4.45 m/s would give 1.9 m, so a 9% speed deficit is sufficient. The next step is to test, not assume.
2. Design a 15-minute investigative sequence for Group B that uses available equipment (video-enabled phone, protractor, photogate, meter stick, spare launcher) to identify the cause. Justify each step by what alternative it rules out.
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- Video the launch at the muzzle with a phone in slow motion and a meter stick in the frame. Measure the ball's displacement over the first few frames and the angle of the velocity vector. This tests whether the photogate reading and the 35° protractor reading describe the actual launch.
- Swap launchers but keep the same setting (or use a second identical launcher). If B's short range follows the launcher, the spring/seating is implicated; if it follows the group's procedure, the ball-seating or measurement is implicated.
- Check seating and photogate alignment. Seat the ball with a marked push rod and confirm the beam passes through the ball's center. Off-center passage makes the blocked chord shorter than the diameter and biases the speed high.
- Launch at 30°, 35°, and 40° and compare each range to R = v0² sin(2θ)/g using the video-measured v0. A persistent vertical offset that scales with angle points to angle error; a uniform speed offset points to the launcher. Record all trials; deleting data is not part of the sequence.
3. How should the teacher respond to the suggestion to delete the farthest trial, and what does this reveal about the purpose of laboratory work?
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Do not delete the farthest trial. Deleting data because it conflicts with a model inverts the purpose of the lab: data are not obstacles to a predetermined answer. An outlier may be removed only when a documented physical cause has been identified (e.g., the ball struck the clamp), and even then the removal and reason should be reported.
Use the moment to teach model–data comparison and uncertainty. Have Group B report the mean and spread, compare the 17% discrepancy with the propagated uncertainty in v0, angle, and range, and state whether the disagreement is significant. Ask what evidence would distinguish a bad launch from a bad model. This is more valuable than a clean result because it mirrors real experimental physics: systematic effects are diagnosed, not deleted.