Answer:
Explanation:
Notation
In order to do the dimensional analysis we need to take in count that we need to conditions:
a) The energy A is released in a small place
b) The shock follows a spherical pattern
We can assume that the size of the explosion r is a function of the time t, and depends of A (energy), the time (t) and the density of the air is constant .
And now we can solve the dimensional problem. We assume that L is for the distance T for the time and M for the mass.
[r]=L with r representing the radius
[A]= A represent the energy and is defined as the mass times the velocity square, and the velocity is defined as
[t]=T represent the time
represent the density.
Solution to the problem
And if we analyze the function for r we got this:
And if we replpace the formulas for each on we got:
And using algebra properties we can express this like that:
And on this case we can use the exponents to solve the values of x, y and z. We have the following system.
We can solve for x like this x=-y and replacing into quation 2 we got:
And then we can solve for x and we got:
And if we solve for z we got:
And now we can express the radius in terms of the dimensional analysis like this:
And K represent a constant in order to make the porportional relation and equality.
The problem says that we can assume the constant K=1.
And if we solve for the energy we got:
And now we can replace the values given. On this case t =0.025 s, the radius r =140 m, and the density is a constant assumed , and replacing we got:
And we can convert this into ergs we got:
And then we know that 1 g of TNT have
And we got: