You decide to treat yourself to Nerf gun because the semester is getting stressf

Question

You decide to treat yourself to Nerf gun because the semester is getting stressful and why not? Your Nerf gun works via spring action: when you pull the bolt back it compresses a spring, which then fires a dart when you pull the trigger. You take your new toy home and set up a target. You stand 3.5 m away and aim horizontally at the center of the target. When you fire your dart, which has a mass of 1.75 g, you notice that it sticks to the target 30 cm below the center. You notice that you have to pull the spring back 4.5 cm to fire the dart. What is the spring constant for the spring in the Nerf Gun? How much force did you apply to pull hack the bolt? If you want to change the spring (but nothing else!) to make sure that the dart gets twice as close to the target, what would the new spring constant need to be?

Explanation / Answer

(A) in vertical,

y = vyi t + ay t^2 /2

- 0.30 = 0 - 9.8 t^2 /2

{ vyi is zero because dart was shot horizontally}

t = 0.25 sec

  
in horizontal,

3.5 = v0 ( 0.25)

v0 = 14.14 m/s

Applying energy conservation,

k (0.045^2)/2 = (1.75 x 10^-3) (14.14^2)/2

k = 172.9 N/m

(b) F = k x = 172.9 x 0.045 = 7.78 N

(c) now y = - 0.15 cm

t = sqrt(2 x 0.15 / 9.8) = 0.175 s

v0 = 3.5 / 0.175 = 20 m/s

k (0.045^2) = (1.75 x 10^-3)(20^2)

k = 345 N/m

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