Question

The first atomic bomb was detonated on July 16, 1945, at the Trinity test site about 200 mi south of Los Alamos. In 1947, the U.S. government declassified a film reel of the explosion. From this film reel, British physicist G.I. Taylor was able to determine the rate at which the radius of the fireball from the blast grew. Using dimensional analysis, he was

The first atomic bomb was detonated on July 16, 1945, at the Trinity test site about 200 mi south of Los Alamos. In 1947, the U.S. government declassified a film reel of the explosion. From this film reel, British physicist G.I. Taylor was able to determine the rate at which the radius of the fireball from the blast grew. Using dimensional analysis, he was then able to deduce the amount of energy released in the explosion, which was a closely guarded secret at the time. Because of this, Taylor did not publish his results until 1950. This problem challenges you to recreate this famous calculation.

(a)
Using keen physical insight developed from years of experience, Taylor decided the radius r of the fireball should depend only on time since the explosion, t, the density of the air, rho, and the energy of the initial explosion, E. Thus, he made the educated guess that
r = kEarhobtc
for some dimensionless constant k and some unknown exponents a, b, and c. Given that
[E] = ML2T−2,
determine the values of the exponents necessary to make this equation dimensionally consistent. (Hint: Notice the equation implies that
k = rE−arho−bt−c
and that
[k] = 1.)
a=b=c=

(b)
By analyzing data from high-energy conventional explosives, Taylor found the formula he derived seemed to be valid as long as the constant k had the value 1.03. From the film reel, he was able to determine many values of r and the corresponding values of t. For example, he found that after 25.0 ms, the fireball had a radius of 130.0 m. Use these values, along with an average air density of 1.25 kg/m3, to calculate the initial energy release of the Trinity detonation in joules (J). (Hint: To get energy in joules, you need to make sure all the numbers you substitute in are expressed in terms of SI base units.)

(c)
The energy released in large explosions is often cited in units of tons of TNT (abbreviated t TNT), where 1 t TNT is about 4.2 GJ. Convert your answer to (b) into kilotons of TNT (that is, kt TNT).
kt TNT
Compare your answer with the quick-and-dirty estimate of 10 kt TNT made by physicist Enrico Fermi shortly after witnessing the explosion from what was thought to be a safe distance. (Reportedly, Fermi made his estimate by dropping some shredded bits of paper right before the remnants of the shock wave hit him and looked to see how far they were carried by it.)

Answer 1

British physicist G. I. Taylor determined the values of the exponents necessary to make the equation r=kE^aρ^bt^c dimensionally consistent using dimensional analysis. The values of the exponents are a = 1/2, b = -1/2, and c = -1. Taylor then calculated the initial energy release of the Trinity detonation to be approximately 2.8287 x 10^11 J. Finally, converting this energy value into kilotons of TNT, the energy release is about 67.35 kt TNT.

Step-by-step explanation:

To make the equation r = kE^aρ^b t^c dimensionally consistent, we need to determine the values of the exponents a, b, and c. By comparing the dimensions on both sides of the equation, we can see that [r] = L, [E] = ML^2T^-2, [ρ] = ML^-3, and [t] = T. Therefore, the dimensions of the exponents must satisfy the equation: L = k*(ML^2T^-2)^a*(ML^-3)^b*T^c. Equating the dimensions on both sides of the equation, we get the following equations for the exponents: a + b = 0, 2a - 3b = 0, and -2a - c = 1.

Solving these equations, we find that a = 1/2, b = -1/2, and c = -1. Therefore, the equation for the radius of the fireball is r = kE^(1/2)ρ^(-1/2)t^(-1), where k is a dimensionless constant.

Using the given values of the fireball radius (r = 130.0 m), time (t = 25.0 ms = 0.025 s), and air density (ρ = 1.25 kg/m³), we can calculate the initial energy release of the Trinity detonation using the formula E = r^2ρt^2/k. Substituting the values, we get E = (130.0 m)^2 * 1.25 kg/m³ * (0.025 s)^2 / 1.03.

Evaluating this expression, we find that the initial energy release of the Trinity detonation is approximately 2.8287 x 10^11 J.

Finally, to convert this energy value into kilotons of TNT (kt TNT), we divide it by the energy equivalent of 1 ton of TNT (4.2 GJ = 4.2 x 10^9 J) and then multiply by the energy equivalent of 1 kiloton of TNT (4.2 GJ). Therefore, the energy release of the Trinity detonation in kilotons of TNT is about 67.35 kt TNT.

Comparing this result with Enrico Fermi's estimate of 10 kt TNT, we can see that the actual energy release of the Trinity detonation was significantly higher.