# Energy and radius of a shock wave

• david22
In summary: The data seems to be following a curve that approximates the R(t) function, but it's not a perfect fit.
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)=1.033E0.2ρ-0.2t0.4 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=(1.033E0.2ρ-0.2t0.4)/t = (1.033E0.2ρ-0.2)/t0.6
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 :)

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.

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

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.

thank you I will do it :)

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

david22 said:
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?

## 1. What is a shock wave?

A shock wave is a type of propagating disturbance that moves through a medium, such as air or water, at a speed faster than the speed of sound in that medium. It is characterized by a sudden and rapid increase in pressure, density, and temperature.

## 2. How is the energy of a shock wave determined?

The energy of a shock wave is determined by the amount of energy that is released during the initial disturbance or explosion that created the shock wave. It is also affected by the medium through which the shock wave travels, as well as the distance it travels.

## 3. What factors affect the radius of a shock wave?

The radius of a shock wave is affected by several factors, including the initial energy of the disturbance, the medium through which it travels, and the distance it travels. Other factors such as atmospheric conditions and obstacles in the shock wave's path can also affect its radius.

## 4. What is the relationship between energy and radius of a shock wave?

The relationship between energy and radius of a shock wave is directly proportional. This means that as the energy of the shock wave increases, its radius also increases. However, the rate of increase in radius may vary depending on other factors, such as the medium through which the shock wave travels.

## 5. Can the energy and radius of a shock wave be controlled?

In some cases, the energy and radius of a shock wave can be controlled. For example, in controlled explosions, the amount of energy released can be carefully calculated to produce a desired radius for the shock wave. However, in natural occurrences such as earthquakes or supernovas, the energy and radius of a shock wave cannot be controlled.

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