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# .
A traveling wave on a taut string is described by the following equation:
1 Answer
The vertical speed of such a point is 100 cm/s.
Explanation:
Vertical position is given by
When
#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
Vertical velocity is computed as
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
Since the velocity we want occurs when
#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
#"speed "= 100 "cm"//"s".#