Spring pulled with one end fixed As the title says, if one end of a spring of mass $m$ is fixed to say,  a wall, and the other one is pulled at a constant velocity $v$ by some external agent, we have to find the kinetic energy of the whole spring as a body.
Now, the book where i saw this question just gave the answer with a hint to consider that this is akin to a mass of $m/3$ being pulled while attached to a massless spring. 
My problem is, isnt this in multivariable calculus? Or have I gone wrong. My education level is high school currently.  So thats why I am facing the problem. Please suggest a solution.
 A: I don't know if you have already done integration in high school but at least here is a solution without using multivariable calc. 
The kinetic energy of the spring with length $L$ is the following
$$E_{kin}=\int_m \frac{u^2}{2} \, \text{d}m$$
where $u$ is the speed of a little mass element d$m$ and the integration is over the mass of the spring. Assuming uniform mass density $\mu = \frac{m}{L}$ such that $m(x)=\mu x \implies \text{d}m = \mu \, \text{d}x$
$$\implies E_{kin}=\int_0^L \frac{u^2 \cdot \mu}{2}  \, \text{d}x  $$
The velocity of a mass element is directly proportional to the velocity of the external agent, assuming that it persistently is pulling with velocity $v$, namely $u=\frac{v\cdot x }{L}$. Notice that this gives the correct boundary conditions at $x=0 \implies u=0$ and $x=L \implies u=v$. You can fairly easy convince yourself that this relation holds.
$$\implies E_{kin} = \frac{\mu}{2} \int_0^L \left( \frac{v\cdot x }{L} \right)^2 \text{d}x = \frac{\mu v^2}{2L^2} \int_0^L x^2 \text{d}x = \left.  \frac{\mu v^2 x^3}{2\cdot 3 L^2}  \right|_0^L =  \frac{\mu v^2 L}{2\cdot 3} = \frac{m}{3} \cdot \frac{v^2}{2}$$
where I used the $\mu = \frac{m}{L}$ in the last step. Notice that this is equal to the kinetic energy of an object with mass $\frac{m}{3}$.
I don't know any other way of getting this equation without using integration and I doubt that one exists because my intuition always says that 3 is a sneaky number, which almost always arises from an integral.
Edit after comments:
 
(For those who cannot read the names of points they are from left to right $A$ $D$ $B$ and $B'$)
I have seen that the formula for $u$ created some confusion so let me elaborate on the formula with an example. Assume that the spring is of the length $|AB|=L$ and the point that we are interested in is the middle of the spring ie the point $D$. If I stretch the string by an amount $AB$ in $t$ seconds ie I double the length ($AB'$) of the spring then the middle of the stretched spring is the point $B$. Although the end point of the spring has travelled a distance $AB$ the midpoint of the string has travelled a distance of $|DB|=L/2$ as can be seen from the (poor) drawing that I made. Thus the speed of the little mass element on the point $D$ is $\frac{v}{2}$ ,where $v=\frac{L}{t}$.
