Suppose a 1 -gram paperclip struck a spaceship at 99 % of...

Suppose a 1 -gram paperclip struck a spaceship at $99 \%$ of the speed of light. Special relativity indicates that the paperclip's energy goes from $\gamma \times m c^{2}$ and drops down to its "rest mass energy" of $m c^{2}$ once it stops moving. a. How much energy is released in the collision? b. How much energy is this in kilotons of TNT?

Suppose a 1 -gram paperclip struck a spaceship at $99 \%$ of the speed of light. Special relativity indicates that the paperclip's energy goes from $\gamma \times m c^{2}$ and drops down to its "rest mass energy" of $m c^{2}$ once it stops moving.
a. How much energy is released in the collision?
b. How much energy is this in kilotons of TNT?

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Key Concepts

Special Relativity and Mass–Energy Equivalence

This concept explains that mass and energy are two forms of the same thing, as encapsulated in Einstein's famous equation E = mc². In the context of high-speed collisions, the total energy of an object is a combination of its rest mass energy and the kinetic energy it has due to its motion. When the object comes to a stop, its energy changes, releasing the kinetic component as part of the energy conversion process.

Lorentz Factor and Relativistic Kinetic Energy

The Lorentz factor, often denoted as ?, arises in the math of special relativity to account for the effects of traveling at speeds close to the speed of light. It modifies the energy expressions so that the total energy of a moving object is ?mc². The excess energy over the rest mass energy, given by (? - 1)mc², represents the kinetic energy, which is relevant when assessing energy release in high-speed impacts.

Energy Unit Conversions

Energy conversions are crucial for interpreting the magnitude of energy release in familiar terms. In physics, energy is often calculated in Joules, but comparing this energy to explosive yields, like kilotons of TNT, requires using a standard conversion factor. This concept helps provide an intuitive grasp of just how powerful the energy release is when described in everyday explosive terms.

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