Find transverse velocity given an equation of displacement

[dUDEonAfORUM]

Homework Statement

A wave pulse on a string is given by D(x) = D[0][/SUB]/(x[2][/SUP]+a[2][/SUP]), where D0 is a constant with units of cm3 and a is a constant with units of meters.
a. If the wave moves along the string at a velocity of v in cm/s, what is the transverse velocity of particles on the string at x = a and t = 0?
b. At time t = 0, what is the maximum transverse speed of particles anywhere on the string?

Homework Equations

Displacement as a function of time is given by D(x,t) = D(x - vt).
The transverse velocity is given by v = ∂D/∂t.

The Attempt at a Solution

I have absolutely no idea how to approach this at all. I tried doing the derivative of D(x), but the answer is incorrect. My incorrect answer was v = -D[0][/SUB]/(2a[2][/SUP]*10000)

 

[kuruman]

Your equations are confusing. Can you post a cleaner form? Please use preview before posting. Thanks.

 

[haruspex]

D(x) = D[0][/SUB]/(x[2][/SUP]+a[2][/SUP]),

Do you mean $\frac{D_0}{x^2+a^2}$?

I tried doing the derivative of D(x)

With respect to what? Remember, it is a velocity that's asked for.

 

[dUDEonAfORUM]

Do you mean $\frac{D_0}{x^2+a^2}$?

Yes, this was what I meant.

With respect to what? Remember, it is a velocity that's asked for.

With respect to time t.

I have already figured a. out I did the derivative of D(x,t). The answer I got and believe is correct is, $\frac{100vD_0}{2(a*0.01)^3}$

I still don't know how to do b. though.

 

[kuruman]

If you don't have specific numbers for D₀ and a, you don't have to convert to consistent units.

I still don't know how to do b. though.

When you did part (a), you found the velocity u(x,t) of a point on the string as a function of time and position. What is u(x,0)? At what value of x is it a maximum?

 

[haruspex]

I have already figured a. out I did the derivative of D(x,t). The answer I got and believe is correct is, $\frac{100vD_0}{2(a*0.01)^3}$

Are you sure about the divisor of 2. Seems to me that cancels out with a 2 from the differentiation.
Edit: my mistake; only one 2 cancels.

I'm dubious about your powers of 10 factor.
To avoid confusion, it would be better to write the general wave equation as y(x,t). D is given as a function of only one variable.

The question is badly posed. Since it specifies units for the constants (very bad form in my book) these unknown constants, a, v and D₀ are dimensionless numbers. Consequently, "x=a" does not mean anything unless the units for x are also stated. I take those to be metres also, to match a.
That being so, y(x,t)=D(x-vt/100), no?

For part b, having found the general equation for ∂y/∂t, you can plug in t=0 to get transverse velocity at time 0 as a function of x. Then you just have to find the max wrt x.

 

[haruspex]

If you don't have specific numbers for D0 and a, you don't have to convert to consistent units.

Please read post #6. Conversion is necessary in this case.

 

[kuruman]

Please read post #6. Conversion is necessary in this case.

I agree that the statement of the problem is in bad form, but I think conversion is necessary only in cases where a numerical result is expected. Stated differently, one can always think of variables "D₀" and "a" in the equation as involving the same units of length as "x" and proceed with the derivation of whatever is required. Then if numbers with units are given, one has to substitute appropriately. I understand that it's OK to provide a hybrid formula (in terms of units) and say
"Given $D(x)=\frac{D_0}{x^2+a^2}$ where $D_0$ is in cm³ and $a$ is in meters." But then what are the units of $x$ in the equation? Are we to assume consistency when a and D₀ are already given as not consistent?

Furthermore, questions (a) and (b) do not specify the units of the expected answer. Yes, the right hand side must be made consistent, but should the conversions be made so that D(x,t) on the left side is in centimeters or in meters? That's not obvious. This problem is not bad only in form, it is just bad. The issue with having hybrid units is that whenever a formula (not to be confused with an equation) is written down, the units of all quantities on the right and left side need to be specified for it to make sense.

 

[haruspex]

conversion is necessary only in cases where a numerical result is expected

No, the velocity is a problem. This can be made clear by defining V as the velocity in m/s, so we have V=v/100.
I will assume x is in m.
y(x,t)=D(x-Vt)=D(x-vt/100)
Upon differentiation wrt t, we get a factor V, hence a factor v/100 in the final answer.

D(x,t) on the left side is in centimeters or in meters?

Yes, it is also unclear what the units are for the displacement (y in my notation). I think that will become a problem in trying to answer part b.

This problem is not bad only in form

I meant that it is poor style to specify units for unknowns, even if done consistently. Unknowns should be abstract: known dimension but independent of units.
This can lead to some surprising forms. E.g. if g is gravitational acceleration then the velocity change after 10 seconds is 10g s. The value for g can then be plugged in in any units you care to use and the result comes out right: g=32f/s² gives 10(32 f/s²) s= 320 f/s.

 

[kuruman]

This can lead to some surprising forms.

Also to some impossible forms. Consider the problem "Two identical charges of q statcoulombs are separated by r meters. Find the repulsive force on one of the charges."