I Mass on a string-harmonic oscillator

[Lushikato]

Hello, I encountered a mass on a string problem in which the mass, moved from the equilibrium, gets a harmonic motion. The catch, however, is that the mass of the string is not neglected. On the lecture, the prof. wanted to calculate, for some reason, the complete kinetic energy of the system:

(1/2)∑m_iv_i² =(1/2)∫v₂dm

where the v_i and m_i are parts of the string and their velocities. After that, he said that v(x) has the next form:

v(x)=A+Bx

Can anyone elaborate on why the velocity has the dependence written above?

 

[person_random_normal]

It's specified in the question so why worry about it

 

[person_random_normal]

It's impossible to model the behavior of the string without knowing some specifications
Like
Young's modulus(and it's variations with velocity, acceleration... Of the string)
However,
Why worry about that as it's altogether a different thing !