A kids BB gun uses a spring to fire little metal BBs at a target. The diameter o

Question

A kids BB gun uses a spring to fire little metal BBs at a target. The diameter of a BB is 4.5 mm. Assume the BBs are made of steel with a density of 8.0 times 10^3 kg/m^3, and that the spring in the gun gets compressed by 1.0cm in order to shoot the BB. a. If the BB exits the muzzle at 115 m/s how much energy was stored in the spring? b. What is the elastic constant of the spring? c. What is the acceleration of the BB while the spring is in the act of firing the BB and is still compressed by 0.5 cm? In the previous problem, how high in the sky will the gun be able to shoot a BB if we ignore air drag (not a realistic thing to do, but I'm asking anyway). Please don't use kinematics to answer this, but rather energy considerations from this chapter.

Explanation / Answer

5. (A) m = volume x density

= (4pi/3) (4.5 x 10^-3 / 2)^3 (8 x 10^3)

= 3.82 x 10^-4 kg

energy stored = m v^2 / 2

= (3.82 x 10^-4) (115^2) / 2

=2.52 J

(B) 2.52 = k (0.01^2) / 2

k = 50480.3 N/m

(c) F = kx = 252.4 N

a = F/m = 6.61 x 10^5 m/s^2

6. Applying energy conservation,

initial PE+ KE = PE +KE at highest point

0 + m v^2 /2 = m g H + 0

H = v^2 / (2 g)

H = 115^2 / (2 x9.8)

H = 674.7 m

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