Cuesta College, San Luis Obispo, CA
Anima Kitty
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A 2008 NERF® N-Strike Recon CS-6 toy gun can be modeled as a compressed horizontal spring that is released to push against a standard 1.3×10–3 kg foam dart, which speeds up from rest to a final speed of 16 m/s.[*][**][***] Ignore drag/friction.
A slightly more massive dart launched from this same Recon CS-6 toy gun will have __________ final speed compared to the standard 1.3×10–3 kg dart.
(A) a slower.
(B) the same.
(C) a faster.
(D) (Not enough information given.)
[*] nerfhaven.com/forums/topic/23927-dart-mass-listing/
[**] nerfguns.net/mods/mod-kits/
[***] instructables.com/id/Streamline-Dart-Mod/
Correct answer (highlight to unhide): (B)
Starting with the energy balance equation:
Wnc = ∆KEtr + ∆PEgrav + ∆PEelas,
where Wnc = 0 (no external gains/losses of mechanical energy), and ∆PEgrav = 0 (as there is no change in elevation of the foam dart as it travels horizontally), such that:
0 = ∆KEtr + ∆PEelas,
0 = (1/2)·m·∆(v2) + (1/2)·k·∆(x2),
0 = (1/2)·m·(vf2 – v02) + (1/2)·k·(xf2 – x02).
With the initial parameters of v0 = 0 (starting from rest) and the final parameter xf = 0 (spring relaxed after releasing the foam dart), then:
0 = (1/2)·m·(vf2 – 02) + (1/2)·k·(02 – x02),
0 = (1/2)·m·vf2 – (1/2)·k·x02,
(k·x02)/m = vf2,
and thus the final velocity of the foam dart is given by:
vf = x0·√(k/m).
Using a more massive foam dart in the same toy gun would increase m in the denominator of the square root quantity (while the spring constant k and the distance x0 that is spring is compressed from equilibrium would be unchanged), and thus this would result in a slower final speed compared to the original less massive foam dart.
Sections 70854, 70855, 73320
Exam code: quiz06rn3T
(A) : 44 students
(B) : 8 students
(C) : 4 students
(D) : 0 students
Success level: 79%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.18
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