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Chapter 35: Problem 67

The explosive yield of the atomic bomb dropped on Hiroshima near the end of World War II was approximately 15.0 kilotons of TNT. One kiloton is about \(4.18 \cdot 10^{12} \mathrm{~J}\) of energy. Find the amount of mass that was converted into energy in this bomb.

Short Answer

Expert verified
Answer: Approximately \(6.97 \cdot 10^{-4}\) kg of mass was converted into energy in the atomic bomb dropped on Hiroshima.

Step by step solution

01

Understand the given information

We are given the explosive yield of the atomic bomb dropped on Hiroshima as 15.0 kilotons of TNT, which is equivalent to 15.0 \(\times 4.18 \cdot 10^{12} \mathrm{~J}\) of energy. We will use this information to find the mass converted into energy.
02

Convert the explosive yield in kilotons to Joules

To find the energy in Joules, we need to multiply the explosive yield in kilotons by the conversion rate provided. That is: Energy (J) = 15.0 kilotons × (\(4.18 \cdot 10^{12}\) J/kiloton) thus, Energy (J) = \(6.27 \cdot 10^{13}\) J
03

Use Einstein's equation to find the mass

Einstein's equation states that the energy of an object is equal to its mass multiplied by the speed of light squared (\(E=mc^2\)). We have the energy (E) and need to find the mass (m). The speed of light (c) is approximately \(3.0 \cdot 10^8\ \mathrm{m/s}\) Rearranging the equation to solve for mass, we get: m = \(\frac{E}{c^2}\) Substituting the values we have m = \(\frac{6.27 \cdot 10^{13} \mathrm{~J}}{(3.0 \cdot 10^8\ \mathrm{m/s})^2}\)
04

Calculate the mass

Now calculate the mass by dividing the energy by the speed of light squared: m = \(\frac{6.27 \cdot 10^{13} \mathrm{~J}}{(3.0 \cdot 10^8\ \mathrm{m/s})^2}\) m = \(\frac{6.27 \cdot 10^{13} \mathrm{~J}}{9.0 \cdot 10^{16}\ \mathrm{m^2/s^2}}\) m ≈ \(6.97 \cdot 10^{-4}\) kg So, about \(6.97 \cdot 10^{-4}\) kg of mass was converted into energy in the atomic bomb dropped on Hiroshima.

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