Max transverse speed on a sinusoidal wave

[ichivictus]

Given the below equation of a sinusoidal wave, find the maximum transverse speed of a point on the string.

y(x,t) = .00325m * sin(70x -3t)


I am brand new to waves and trying to figure out what this question exactly means. The way I see it is that it might be at a max speed when 70x-3t = 1 because then the equation would just equal the amplitude.

x is related to distance and t is related to time. the function is a position function I think so taking the derivative should give its velocity function.

dy/dt = v = .2275m cos(70x-3t)

Now my thinking has changed. Would velocity be at its max when 70x-3t = 0 since cos(0) = 1?

70x = 3t
t = 70x/3 or x = 3t/70

Ha, I have no idea where I'm going with this. I'm just studying ahead.

 

[paisiello2]

You did not calculate dy/dt correctly.

The question did not ask you to calculate when or where the maximum speed occurs, only what the maximum speed is.

 

[ichivictus]

Hmmm. How would I take the derivative in this case? Do I even have to take the derivative?

Oh and wouldn't max speed be inbetween 2 amplitudes? Like if you look at a sin(x) graph it would be where y=0. Wouldn't this be where max speed would occur?

 

[ehild]

You did the derivative with respect to x instead of t.

ehild

 

[ichivictus]

Ah I see. So then dy/dt = v = -.00975 cos(70x-3t)

At its max, cos(70x-3t) = 1/2 since that will be inbetween two amplitudes where its Vmax is at (correct me if I'm wrong). This speed will be the same for any x/t combination that works with this idea.

If cos(70x-3t) = 1/2 then v = -.00975m/s * .5 = -.004875 m/s

Since it is negative, I have a hard time believing I'm on the right track. I think it may actually be when cos(...)=1 and it would be the absolute value of that answer. Or maybe when cos(..)= -1 then it would just negate the negative sign.

So then the max speed would actually be .975 cm/s

 

[paisiello2]

The latter answer is correct.

 

[AlephZero]

cos(.) varies between -1 and +1. If you are not sure about that, you need to review your trig courses.

Pick the value in that range which gives you the maximum value for the velocity.

I can't guess where your idea that "At its max,cos(70x-3t) = 1/2" came from.

 

[ichivictus]

cos(.) varies between -1 and +1. If you are not sure about that, you need to review your trig courses.

Pick the value in that range which gives you the maximum value for the velocity.

I can't guess where your idea that "At its max,cos(70x-3t) = 1/2" came from.

I think I was just stuck thinking about oscillations. From what I remember, when y=0 it's at a max velocity and at the top or bottom of its amplitudes, it's at its max acceleration. Something like that.

Thanks though.