To determine the amount of mass that needs to be converted into energy to produce 24.0 megajoules (MJ) of energy, we can use Einstein's famous equation from the theory of relativity:
[tex]\[ E = mc^2 \][/tex]
where:
- [tex]\( E \)[/tex] is the energy produced,
- [tex]\( m \)[/tex] is the mass,
- [tex]\( c \)[/tex] is the speed of light in a vacuum, which is approximately [tex]\( 3 \times 10^8 \)[/tex] meters per second.
Let's break down the steps to find the required mass:
1. Convert the energy from megajoules to joules:
Since 1 megajoule (MJ) equals [tex]\( 1 \times 10^6 \)[/tex] joules (J),
[tex]\[ 24.0 \text{ MJ} = 24.0 \times 10^6 \text{ J} = 24,000,000 \text{ J} \][/tex]
2. Use the equation [tex]\( E = mc^2 \)[/tex] to solve for mass [tex]\( m \)[/tex]:
Rearrange the equation to solve for [tex]\( m \)[/tex]:
[tex]\[ m = \frac{E}{c^2} \][/tex]
3. Substitute the values of [tex]\( E \)[/tex] (24,000,000 J) and [tex]\( c \)[/tex] ([tex]\( 3 \times 10^8 \)[/tex] m/s):
[tex]\[ m = \frac{24,000,000 \text{ J}}{(3 \times 10^8 \text{ m/s})^2} \][/tex]
[tex]\[ m = \frac{24,000,000 \text{ J}}{9 \times 10^{16} \text{ m}^2/\text{s}^2} \][/tex]
4. Perform the division:
[tex]\[ m = \frac{24,000,000}{9 \times 10^{16}} \][/tex]
[tex]\[ m = \frac{24}{9} \times 10^{-10} \][/tex]
[tex]\[ m = \frac{8}{3} \times 10^{-10} \][/tex]
5. Simplify the result:
[tex]\[ m \approx 2.67 \times 10^{-10} \text{ kg} \][/tex]
Therefore, approximately [tex]\( 2.67 \times 10^{-10} \)[/tex] kilograms of matter would need to be converted into energy to produce 24.0 megajoules of energy.
