Marble Track Range Inconsistent with Energy Conservation
A student releases a 20 g steel marble from rest at point A, 1.20 m above the floor, on a curved track. The track ends horizontally at point B, 0.80 m above the floor. The marble leaves B and lands on the floor a horizontal distance of 1.50 m from B. The student assumes the track is frictionless, uses the measured range to calculate the exit speed, and compares it with the speed predicted by energy conservation from A to B. The predicted drop is only 0.40 m, so the student is surprised that the measured range is so large. The marble is small, the track is smooth but not polished, and the landing point is measured with a metre rule to ±1 cm. The student asks whether the track must have a hidden spring or whether the experiment can be trusted.
1. Calculate the exit speed from the measured range and the speed expected if the track is frictionless. State whether the two values agree.
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For a horizontal launch from height h=0.80 m, the time of flight is t=sqrt(2h/g)=sqrt(1.60/9.8)=0.404 s. The horizontal exit speed is v=x/t=1.50/0.404=3.71 m/s. If the track is frictionless and the marble starts from rest, energy conservation gives (1/2)mv^2=mg(1.20−0.80), so v=sqrt(2×9.8×0.40)=sqrt(7.84)=2.80 m/s. The measured speed is about 33% larger, not smaller. Friction can only remove mechanical energy, so a smooth track cannot explain a larger speed. Either the geometry is wrong, the range measurement is wrong, or some source added energy. The frictionless assumption is inconsistent with the data.
2. Identify two plausible sources of the inconsistency that are not a hidden spring, and describe one measurement that would distinguish them.
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One possibility is that the release height is larger than 1.20 m, for example if the metre rule was referenced to the track surface rather than the floor, or if the marble was pushed. Another is that the exit height is lower than 0.80 m, or that the landing distance was measured to the first bounce rather than to the floor. A direct check is to measure the speed at B with a light gate or high-speed video. If the gate gives about 2.8 m/s, the launch geometry or range measurement is at fault; if it gives about 3.7 m/s, energy was added by a push, a hidden spring, or a lower release point. Repeating with a straight, level exit and a plumb line improves the height measurement.
3. If the steel marble is replaced by a hollow plastic ball of the same size and mass, how would the landing distance change qualitatively, and why?
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For a rolling ball, gravitational potential energy splits into translational and rotational kinetic energy. A hollow ball has a larger moment of inertia for its radius than a solid steel marble, so a larger fraction of the available energy goes into rotation and the translational exit speed is smaller. With the same release height and track, the hollow ball should therefore leave B more slowly and land closer, assuming rolling without slipping dominates. If the track is truly frictionless and the ball slides, there is no rotational energy and the range would be essentially unchanged; the comparison is therefore also a test of whether rolling is occurring.