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A dart is loaded into a spring-loaded toy dart gun by pushing the spring in by a 
distance d. For the next loading?the spring is compressed a distance d/3. How much 
work is required to load the second dart compared to that required to load the first?A) nine times as muchB) three times as muchC) the sameD) one-third as muchE) one-Ninth as much

Respuesta :

Answer:

[tex]W'=\dfrac{W}{9}[/tex]

Explanation:

The potential energy of the spring or the work done by the spring is given by :

[tex]W=\dfrac{1}{2}kd^2[/tex]............(1)

k is the spring constant

d is the compression

When the spring is compressed a distance d' = d/3, let W' is the work is required to load the second dart. Then the work done is given by :

[tex]W'=\dfrac{1}{2}kd'^2[/tex]

[tex]W'=\dfrac{1}{2}k(d/3)^2[/tex].............(2)

Dividing equation (1) and (2) :

[tex]\dfrac{W}{W'}=\dfrac{1/2kd^2}{1/2k(d/3)^2}[/tex]

[tex]\dfrac{W}{W'}=9[/tex]

[tex]W'=\dfrac{W}{9}[/tex]

So, the work required  to load the second dart compared to that required to load the first is one-Ninth as much. Therefore, the correct option is (E).