Spring-Launcher Range Data and Error Analysis
Three lab groups use identical spring-loaded launchers to shoot a steel ball horizontally from a table 1.20 m high. They measure horizontal range for compression distances x = 2.0, 4.0, and 6.0 cm. Group 1 obtains 0.78, 1.55, and 2.33 m; Group 2 obtains 0.40, 0.82, and 1.18 m; Group 3 obtains 0.74, 1.90, and 2.30 m. The launcher spring constant is listed as 180 N/m, and the ball used by Groups 1 and 3 has mass 24 g. Group 2 used a different ball of the same diameter but unknown mass. Group 2 claims their data show that range is proportional to x and therefore the spring force is constant. Group 1 claims their data prove energy is conserved. The teacher notes that the table edge is not sharp, so the ball may contact the table after launch, and that the launcher clamp was loose during one group's trials.
1. Evaluate Group 2's claim that range is proportional to x and therefore the spring force is constant.
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Group 2's ranges roughly triple as x triples (0.40 to 1.18 m is a factor of 2.95), so range is approximately proportional to x. But that does not mean the force is constant. For an ideal spring, elastic potential energy is 1/2 kx^2, so launch speed v = sqrt(k/m)x and range R = v sqrt(2h/g), giving R proportional to x. A constant force would give energy Fx and v proportional to sqrt(x), so range would scale as sqrt(x), not x. The linear pattern therefore supports a spring force that increases with compression, F = kx, not a constant force. Group 2's smaller absolute ranges are consistent with a heavier ball, since v is proportional to 1/sqrt(m), but the mass is unknown and must be measured before comparing groups.
2. Which group's data are most consistent with ideal behavior, and what could explain Group 3's pattern?
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Group 1 is most consistent with ideal projectile motion from a spring launcher: doubling x from 2.0 to 4.0 cm increases range from 0.78 to 1.55 m (1.99 times), and tripling x gives 2.33 m (2.99 times). Group 2 is also approximately linear but with a lower slope, which a heavier ball would produce. Group 3 is not linear: 1.90 m is 2.57 times the 2.0 cm range, and 2.30 m is only 1.21 times the 4.0 cm range. A different mass changes slope, not shape, so mass alone does not explain Group 3. Plausible causes are the ball striking the table edge after launch, the loose clamp allowing recoil, inconsistent compression measurement, or the spring not returning to the same equilibrium. Repeated trials and a direct launch-speed measurement would distinguish these.
3. Propose two improvements and state what each controls.
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First, clamp the launcher rigidly and verify that the launch direction is horizontal; this controls recoil and keeps the launch angle and height constant. Second, use the same ball for all groups and measure its mass; this controls the mass-dependent launch speed and makes the slopes comparable. Additional improvements: sharpen or extend the table edge so the ball clears it cleanly, measure compression with a fixed scale rather than a ruler held by hand, take at least five trials per compression and report means and ranges, and use two photogates to measure launch speed directly. The photogates eliminate assumptions about the projectile motion and table contact. Repeating with a known spring constant or a force sensor would check whether the spring behaves ideally.