How Does a Traveling Pulse Behave on a String?

[Blairo28]

Homework Statement

The displacement of a pulse traveling on a horizontal string is approximated by:

y(x,t) = 0.05/((1.5x - 3t)^2)+1

where x and y are measured by metres.

i) the velocity of propigation
ii) the direction of motion
iii) the maximum displacement
iv) the displacement at x = 4m after t = 2.5
v)the particle velocity (magnitude and direction) at x = 4 after 2.5s

Homework Equations

Unsure

The Attempt at a Solution

My problem is slightly in understanding the question.
i) Well I think the velocity is given by the change in x over the change in t. Is that what is being asked? I am not sure.

ii) I think that the wave is going in the positive direction due to the negative in (1.5x - 3t) since I read that in the notes but that was using an equation that wasnt squared or having a value on top of it.

iii) Will the just be 0.05/1 so when x and t = 0.

iv) For this part I think I am just plugging those values into the equation right?

v) I am very unsure about this part.

Help with any parts would be appriciated!

 

[anathema]

Hello! I am a scientist and I can help you with this question.

i) Yes, the velocity of propagation can be calculated by finding the change in x over the change in t. In this case, it would be the coefficient of x (1.5) divided by the coefficient of t (3), which gives a velocity of 0.5 meters per second.

ii) The direction of motion can be determined by looking at the sign of the coefficient of t. Since it is negative (-3), the pulse is moving in the positive direction.

iii) The maximum displacement can be found by setting x and t equal to 0, which would give a maximum displacement of 1 meter.

iv) Yes, for this part, you can plug in the given values of x and t into the equation to find the displacement at that specific point in time and space.

v) To find the particle velocity at x=4 after 2.5 seconds, you can first calculate the derivative of the equation with respect to t. This would give you the expression for the particle velocity, which is 0.1/((1.5x - 3t)^3). Then, you can plug in the given values of x and t to find the magnitude and direction of the particle velocity at that point.