# How can we calculate the speed from $E=mc^2$? [closed]

how can we calculate the speed of an object using the formula of Einstein $E=mc^2$?

• In special relativity, $E=\gamma m c^2$. In your formula $\gamma=1$, so the speed is zero. – safesphere Jun 2 '18 at 18:31
• @safesphere so what is the equation to find the speed ? – Taher Jun 2 '18 at 19:04
• $v=c\sqrt{1-\dfrac{1}{\gamma^2}}$ where $\gamma=\dfrac{m}{m_o}$. Thus $v=c\sqrt{1-\left(\dfrac{m_o}{m}\right)^2}$ – safesphere Jun 2 '18 at 20:57

To find the speed, you need the formula for the kinetic energy, which is

$$KE = m_0c^2\left(\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}-1\right)$$

where $m_0c^2$ is the rest mass energy. Solve for $v$.

• can you simplify little bit more ? i mean v = ? – Taher Jun 2 '18 at 19:14
• You can't do that algebra? – HiddenBabel Jun 2 '18 at 19:47
• i mean how do we calculate m0c2 – Taher Jun 2 '18 at 20:05
• $m_0$ is dependent on the object. If you have an electron, you know that $m_0c^2 = 0.511\;\text{MeV}$. If you have a macroscopic object like a baseball, $m_0$ is simply the mass you measure while it's at rest here on Earth. – HiddenBabel Jun 2 '18 at 20:12

In $E = mc^2$, $E$ is the total energy of the object, including chemical, kinetic, thermal, zero point energy, and doubtless a few others. (Zero point energy - NOT Syndrome's gloves, but the energy that something has when it's static at a temperature of 0 K.)

So, if you know the rest mass of the object, $m_0$ - what its mass would be if it currently had zero velocity, but had the same energy in all other forms - and its current mass, $m$, then you could calculate its kinetic energy as $KE = mc^2 - m_0c^2$. If the estimate $KE = \frac 12 mv^2$ indicates that $v \ll c$, then you can use the Newtonian kinetic energy formula to solve for $v$. If, however, $v$ is a significant fraction of $c$, then you will need to read up on Special Relativity - or just click the link below to HyperPhysics and try one situation at a time.

Referred to HyperPhysics.

• what is the rest mass m0c2 ? – Taher Jun 2 '18 at 20:07
• @Taher - the rest mass $m_0$ is the object's mass when it is at rest - when $v=0$. Relativity says that the effective mass will increase as energy increases, so it will take more force to accelerate by $1 \frac ms$ when it is moving $10,000 \frac ms$ than when it is moving $1 \frac ms$. – Post169 Jun 2 '18 at 20:16

The equation $E=mc^2$ is only valid in the rest frame. Thus the speed is always zero.

In the comment thread you asked what is the correct equation to calculate the speed. It is impossible to answer without knowing which quantities you already know. Without considering that, it could be any equation that involves speed.

Sorry for posting this comment as an “answer”, the app won’t let me comment directly.

• Posting comments to questions you didn't ask is on SE is a privilege you earn after you get when you accumulate 50 points on Physics SE. Please read What is Reputation and how do I earn and lose it ?. Please don't try and leapfrog this rule - it's there for a reason. – StephenG Jun 2 '18 at 20:09