7 In a dart gun (see fig. 6.32 on page 219 of the textbook), a spring with the s

Question

7

In a dart gun (see fig. 6.32 on page 219 of the textbook), a spring with the spring constant k=452.9 N/m is compressed 12.1 cm when the 16.8-g dart is loaded. The spring is released, but gets caught inside the gun, stopping at the point where it is still compressed by 3.4 cm. The dart is not caught inside the gun, but is released.

Find the muzzle speed of the dart.

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CHEGG,

Please start with (show how you derive from) simple/basic formulas, and please clearly label and show ALL your work. Thank you!

Before (compressed spring) After (relaxed spring) vi = 0 xi =-80 cm Figure 6.32 Dart gun (a) before and (b) after firing. The spring was compressed by 8.0 cm when the gun was cocked.

Explanation / Answer

potential energy = k x^2 /2

as x (compression ) decreases, spring's PE get converts into KE of bullet.

Loss in PE = 452.9 ( 0.121^2 - 0.034^2 ) /2 = 3.05 J

so using energy conservation,

gain in KE = loss in PE = 3.05 J

KE = m v^2 / 2

3.05 = 0.0168 v^2 / 2

v = 19.06 m/s

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