Energy and radius of a shock wave

[david22]

Homework Statement

A shock wave moves away from the center of the explosion, its pressure is decreasing, and its speed tends to a constant value. In the filming of a particular explosion, the following data was obtained:
t(s): 0, 0.02, 0.04, 0.06 ,0.08, 0.1, 0.12, 0.14, 0.16, 0.18, 0.2, 0.22, 0.24, 0.26
R(m):0, 29.4, 45.1, 59.9, 61.7, 72.5, 76.6, 84.4 , 93.3, 103.9, 108.7, 115.6, 121.1, 129.4
where t(s) is time in seconds and R(m) is the radius of the explosion in meters(for example at 0.2 secons the radius of the explosion is 108.7 meters), the questions are:
a)Using the data, determine the time interval in which the relation for R(t)(see below) is valid
b)Determine the final speed of the shock wave
c)If the shock wave is produced at floor level, then the hemispheric wave would have double energy due to the reflection on the floor than the one that is spherical. Using the above data, determine the total energy released in the explosion

Homework Equations

the relation for R(t) is : R(t)=1.033E^0.2ρ^-0.2t^0.4 where E is the energy released in the explosion, ρ is the density of air, t is the time and R(t) is the radius

The Attempt at a Solution

For b) I divided the relation by t: R(t)/t=(1.033E^0.2ρ^-0.2t^0.4)/t = (1.033E^0.2ρ^-0.2)/t^0.6
Then I took the limit of the last expression when t tends to infinity an that limit is 0, so I found that the final speed is 0. Is this correct?

For a) and For c) I don't know how can I find the interval of time in which the relation is valid I don't know how to determine the energy of the explosion, I can´t think of anything so I really would appreciate if you can help me whith the problem, thanks a lot :)

 

[mfb]

Did you plot the data and the function for different values of E?

(b) If the shock wave would follow this formula forever, the limit would be 0, indeed. But if you do (a) first, you should see that this formula is a bad approximation for large t.

 

[david22]

so for a) i just have to plot different values for the energy and see what happens?

 

[mfb]

That is a good way to start. It should show you how to solve (a) and (b).

If you have data and you are unsure what to do, always plot it.

 

[david22]

thank you I will do it :)

 

[david22]

One last question why is taking the limit when t tends to infinity wrong? because the data just gives me the radius of the shock wave for small values of t

 

[haruspex]

One last question why is taking the limit when t tends to infinity wrong? because the data just gives me the radius of the shock wave for small values of t

It tells you that the R(t) formula is only valid initially, so it certainly cannot be trusted to give the right limit at infinity.
Compare the data with a curve obtained from the R(t) formula. OK, you don't know E and ρ, but they're just constants, so look at the ratio between the observed data and t^0.4. Where the formula is valid, what would you expect to see?
It also tells you that the speed tends to a constant; when you look at a graph of the expansion, does it look like it has become a straight line near the end?