Question

Please Answer?

A traveling wave on a taut string is described by the following equation: `y(x,t)=13"cm"cos(x−3πt)`. Find the speed `("in cm/s")` of a point on the string when its position in the y direction is `7 cm`.

Answer 1

The vertical speed of such a point is 100 cm/s.

Explanation:

Vertical position is given by `y(x, t).`

When `y(x, t)="7 cm",` we get

`y(x,t)="13 cm"cos(x-3pit)`
`"7 cm   " ="13 cm"cos(x-3pit)`
`7/13=cos(x-3pit)`
`cos^(–1)(7/13)=x-3pit`

So `x-3pit=1.00 " rad"`. Of course, there are multiple locations along this wave that, at any given displacement `x` and time `t`, will have vertical position `y="7 cm"`, but they'll all give the same vertical speed.

Vertical velocity is computed as `d/(dt)y(x,t).` Since we are asked for the speed, which disregards direction (+/-), we will take the absolute value of whatever velocity we get.

Calculate:

`d/(dt)y(x,t)="13cm" (–3pi"Hz")[–sin(x-3pit)]`
`color(white)(d/(dt)y(x,t))=(39pi" cm"//"s") sin(x-3pit)`

Remember, the units of `3pit` are `"rad"//"s"` (i.e. Hz), so when we differentiate with respect to `t`, the Hz comes out with the `3pi`.

Since the velocity we want occurs when `x-3pit=1.00,` we substitute:

`d/(dt)y(x,t)=(39pi" cm"//"s")sin("1.00")`
`color(white)(d/(dt)y(x,t))=(39pi" cm"//"s")(0.84265)`
`color(white)(d/(dt)y(x,t))=103.24" cm"//"s"`

The absolute value of this is also 103.24 cm/s. Finally, we were given a `y` displacement of 7 cm, which has one significant figure, so our answer should as well:

`"speed "= 100 "cm"//"s".`