A supernova explosion of a 2.00 times 10 to the 31st kilogram star produces 1.00 times 10 to the 44th joules of energy. How many kilograms of mass are converted to energy in the explosion? What is the ratio Δ𝑚 to 𝑚 of mass destroyed to the original mass of the star?
We’re told in this statement that the star has an initial mass of 2.00 times 10 to the 31st kilograms; we’ll call that value 𝑚. We’re told that the supernova explosion produces 1.00 times 10 to the 44th joules of energy, an amount we’ll call 𝐸. We want to know the number of kilograms of mass converted to energy, which we’ll call Δ𝑚. We also want to know the ratio of mass destroyed to the original mass.
In part two, the ratio Δ𝑚 to 𝑚 of mass destroyed to the original mass of the star we can write as Δ𝑚 divided by 𝑚 to 𝑚, the original star’s mass. To start on our solution, we can recall that energy and mass are equivalent to one another, related by Einstein’s famous equation 𝐸 equals 𝑚𝑐 squared.
When we apply this relationship to our scenario, it tells us that Δ𝑚 is equal to 𝐸, the energy of the supernova explosion, divided by 𝑐 squared, where 𝑐 is the speed of light, which we’ll assume to be exactly equal to 3.00 times 10 to the eighth meters per second.
We’re now ready to plug in for 𝐸 and 𝑐. When we do and we calculate this fraction, we find that Δ𝑚 is 1.11 times 10 to the 27th kilograms. That’s the amount of mass converted to energy in this explosion.
We’re now ready to solve for the ratio Δ𝑚 to 𝑚 to the original mass of the star 𝑚. The change in mass, which we solved for in part one, divided by the original mass of the planet is equal to three significant figures to 5.56 times 10 to the negative fifth.
This fractional change in mass related to the original mass 𝑚 of the planet is therefore 5.56 times 10 to the negative fifth to one. That’s the ratio of the change in mass of the planet to its original mass.