Question

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

Answer 1

Answer:

$W'=\dfrac{W}{9}$

Explanation:

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

$W=\dfrac{1}{2}kd^2$............(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 :

$W'=\dfrac{1}{2}kd'^2$

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

Dividing equation (1) and (2) :

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

$\dfrac{W}{W'}=9$

$W'=\dfrac{W}{9}$

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).