Physics quiz question: foam dart kinetic energy

Physics 205A Quiz 6, fall semester 2016
Cuesta College, San Luis Obispo, CA

"Nerf Recon CS-6"
Anima Kitty
flic.kr/p/4meXDZ

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 a Recon CS-6 will have __________ translational kinetic energy compared to the standard 1.3×10^–3 kg dart.[*][**][***]
(A) less.
(B) the same amount of.
(C) more.
(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:

W_nc = ∆KE_tr + ∆PE_grav + ∆PE_elas,

where W_nc = 0 (no external gains/losses of mechanical energy), and ∆PE_grav = 0 (as there is no change in elevation of the foam dart as it travels horizontally), such that:

0 = ∆KE_tr + ∆PE_elas,

0 = (1/2)·m·∆(v²) + (1/2)·k·∆(x²),

0 = (1/2)·(1.3×10^–3 kg)·(v_f² – v₀²) + (1/2)·k·(x_f² – x₀²).

With initial parameters of v₀ = 0 (starting from rest) and x₀ ≠ 0 (compressed a certain distance from equilibrium), and final parameters v_f = 16 m/s and x_f = 0 (spring relaxed after releasing the foam dart), then:

0 = (1/2)·(1.3×10^–3 kg)·((16 m/s)² – 0²) + (1/2)·k·(0² – x₀²),

0 = +0.1664 J – (1/2)·k·x₀²,

where the quantity +0.1664 J (or to two significant figures, 0.17 J) represents the increase in translational kinetic energy of the foam dart while it is launched, and the second term on the right-hand side of the equation is the decrease in the elastic potential energy of the spring as the foam dart is launched. This means that the spring, when fully compressed, stores 0.17 J of elastic potential energy that is subsequently given to the translational kinetic energy of the foam dart.

Using a more massive foam dart in the same toy gun would neither change the spring constant k nor the distance x₀ that is spring is compressed from equilibrium, and thus this more massive foam dart will still receive the same amount of translational kinetic energy (0.17 J) released from the elastic potential energy of the spring as before.

Sections 70854, 70855, 73320
Exam code: quiz06rn3T
(A) : 10 students
(B) : 17 students
(C) : 28 students
(D) : 0 students

Success level: 30%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.38