Stretching of massive spring and it's kinetic energy

While deriving the expression for the kinetic energy possessed by a massive spring (of mass $m$ and length $l$) when one of the ends is pulled constantly by a velocity $v_0$ while the other end being fixed to a wall (or simply rendered motionless), my physics teacher used following arguments/statements.

If spring stretches uniformly
$frac{v}{x} = frac{v_0}{l} $($v$ being the velocity of an infinitesimally small element of the spring at a distance $ x$)
$ Rightarrow v =(frac{v_0}{l})x$
With this, the kinetic energy of spring becomes
$K_{spring} = int_0^l frac{1}{2} (frac{m}{l}) {rm d}x frac{v^2}{l^2}x^2 {rm Thus, space} K_{spring}= frac{1}{6} mv_0^2 $

My problem is, if the spring is stretching uniformly, how can we say that $frac{v}{x} = frac{v_0}{l}$. I’ve trouble visualising the situation.

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