Math solution on Mass & energy

The relationship between mass and energy is most famously expressed by Einstein's equation from his theory of relativity:

$$𝐸=𝑚{𝑐}^2$$

where:

• $𝐸$ is the energy,
• $𝑚$ is the mass,
• $𝑐$ is the speed of light in a vacuum ($𝑐\approx 3\times 10^8\text{m/s}$).

This equation shows that mass can be converted into energy and vice versa. It has profound implications in fields such as nuclear physics, where small amounts of mass can be converted into large amounts of energy.

Example Problem

Let's solve an example problem step-by-step using this formula.

Problem

Calculate the energy equivalent of 1 gram of mass.

Solution

1. Identify the given values:

   • Mass, $𝑚=1\text{g}=1\times 10^{-3}\text{kg}$ (since $1\text{g}=0.001\text{kg}$)
   • Speed of light, $𝑐=3\times 10^8\text{m/s}$

2. Substitute the values into the mass-energy equivalence formula:

   $$𝐸=𝑚{𝑐}^2$$
   $$𝐸=(1\times 10^{-3}\text{kg})\times (3\times 10^8\text{m/s}{)}^2$$

3. Calculate the term ${𝑐}^2$:

   $${𝑐}^2=(3\times 10^8\text{m/s}{)}^2=9\times 10^{16}{\text{m}}^2\mathrm{/}{\text{s}}^2$$

4. Continue with the calculation:

   $$𝐸=(1\times 10^{-3}\text{kg})\times (9\times 10^{16}{\text{m}}^2\mathrm{/}{\text{s}}^2)$$
   $$𝐸=9\times 10^{13}\text{kg}\cdot {\text{m}}^2\mathrm{/}{\text{s}}^2=9\times 10^{13}\text{J}$$

So, the energy equivalent of 1 gram of mass is $9\times 10^{13}\text{joules}$.

Conclusion

Using Einstein's mass-energy equivalence formula, we determined that 1 gram of mass is equivalent to $9\times 10^{13}\text{joules}$ of energy. This enormous amount of energy from a small mass exemplifies why nuclear reactions, which convert mass into energy, can release such tremendous amounts of energy.