Finding Time for a Wave on a Rope

[zorro]

Homework Statement

A uniform rope of mass m hangs vertically from the ceiling, with its lower end free. A disturbance on the rope travels upward from the lower end. Find the time taken by the disturbance to reach the top of the rope if the length of the rope is L.

The Attempt at a Solution

Due to the effect of gravity, velocity of the wave will decrease as it travels up.
v at a distance x from the free end is given by v = $$\sqrt{2gx}$$

dt= dx/v
substituting for v and then integrating with limits 0 to t and 0 to L resp we get
t = $$\sqrt{2L/g}$$

The answer is t = 2$$\sqrt{L/g}$$

 

[Doc Al]

v at a distance x from the free end is given by v = $$\sqrt{2gx}$$

How did you arrive at this?

 

[zorro]

By using equation of motion v² - u² = 2as
I figured out that I used u = 0 which is wrong.
How do we find the velocity then?

 

[Doc Al]

By using equation of motion v² - u² = 2as
I figured out that I used u = 0 which is wrong.

That's a kinematic equation for motion under constant acceleration; not relevant here.

How do we find the velocity then?

What properties of the string determine the speed of a wave?

 

[zorro]

It depends on the elastic and intertial properties of the material of string.
V is given by √(TL/M) where M is the mass of the string and L is its length
At a distance x from the bottom, tension is Mgx/L
Substituting this in equation I got the correct answer.
Thanks alot!