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Chapter 12: Problem 9

Does \(E_{0}=m c^{2}\) apply only to objects traveling at the speed of light? Why or why not?

Short Answer

Expert verified
The equation \(E_0 = mc^2\) does not apply only to objects traveling at the speed of light. It represents the rest energy (\(E_0\)) of any object, regardless of its velocity. The relativistic energy of an object can be derived from the theory of special relativity and is given by the expression \(E = \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}}\). This equation is valid for objects with velocities less than, equal to, or approaching the speed of light.

Step by step solution

01

Understanding the equation

The equation \(E_0 = mc^2\) is known as the mass-energy equivalence, which states that the energy (E) of an object is equal to its mass (m) multiplied by the speed of light (c) squared.
02

Deriving the expression for the relativistic energy

The relativistic energy (\(E\)) of an object can be derived from the theory of special relativity: \(E^2 = (mc^2)^2 + (pc)^2\), where \(p\) represents the momentum of the object and is given by \(p = m \cdot v / \sqrt{1 - \frac{v^2}{c^2}}\).
03

Evaluating the relativistic energy at the speed of light

As \(v\) approaches the speed of light (\(c\)), the denominator of the momentum equation approaches zero, and the momentum (\(p\)) approaches infinity. Therefore, the \$E^2 = (mc^2)^2 + (pc)^2\$ equation can be rewritten as \(E = mc^2\) when \(v = c\).
04

Evaluating the equation for other velocities

If we substitute the momentum expression into the relativistic energy equation, we can see its behavior for objects at velocities other than the speed of light. For \(v = 0\) (stationary object), \(E = mc^2\). This represents the rest energy of an object (\(E_0\)). For \(0 \le v \ne c\), \(E = \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}}\). Notice that the equation still gives us a finite value for energy even when the object is traveling at a velocity that is not equal to the speed of light. In conclusion, \(E_0 = mc^2\) does not apply only to objects traveling at the speed of light. Instead, it represents the rest energy (\(E_0\)) of any object regardless of its velocity.

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