Question

In physics, energy E carries dimensions of mass times length squared, divided by time squared. Use dimensional
analysis to derive a relationship for energy in terms of mass m and speed v, up to a constant of proportionality. Set
the speed equal to c, the speed of light, and the constant of proportionality equal to 1 to get the most famous equation
in physics.

Answer 1

Let's describe energy in terms of the dimensions described:\[[Energy]=[Mass] \times [Length]^2 \times [Time]^{-2}\]Now that we must express energy in terms of mass and speed, we must show the dimensions of velocity to see how it fits into energy:\[[Speed]=\frac{[Length]}{[Time]}\]Now if we look back at the dimensions of energy, we see that we can rewrite it a bit to coincide better with this definition of velocity:\[[Energy]=[Mass] \times \frac{[Length]^2}{[Time]^2} \rightarrow [Mass] \times (\frac{[Length]}{[Time]})^2\]Since...\[[Speed]=\frac{[Length]}{[Time]}\]We can substitute this into our energy definition:\[[Energy]=[Mass] \times [Speed]^2\]Knowing that the symbol for energy is E, the symbol for mass, m, and the symbol for the speed of light, c, can you derive what equation the problem is asking for?

Answer 2

nice! just wanted to make sure, thanks.
The derived equation would be: \[E=mc ^{2}\] right?

Answer 3

That's right!