Find the kinetic energy of a 78.0-kg spacecraft launched out of the solar system with speed 106

step 1 Hi everyone here we have to calculate kinetic energy of 78 KG spacecraft launched out of the sol

Find the kinetic energy of a 78.0-kg spacecraft launched out...

Find the kinetic energy of a $78.0-\mathrm{kg}$ spacecraft launched out of the solar system with speed $106 \mathrm{km} / \mathrm{s}$ by using (a) the classical equation $K=\frac{1}{2} m u^{2}$. (b) What If? Calculate its kinetic energy using the relativistic equation.

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Key Concepts

Classical Kinetic Energy

Classical kinetic energy represents the energy due to motion as quantified by the formula K = 1/2 m v², where m is mass and v is velocity. This expression is derived under the assumptions of Newtonian mechanics, making it highly accurate when the speed of an object is much less than the speed of light, so that relativistic corrections are negligible.

Relativistic Kinetic Energy

Relativistic kinetic energy extends the classical concept by incorporating the effects of special relativity, which become significant as an object's speed approaches the speed of light. It is expressed as K = (? - 1) m c², where ? is the Lorentz factor. This formulation accounts for the fact that mass and energy are interrelated, and it predicts deviations from classical predictions at high speeds.

Lorentz Factor

The Lorentz factor, denoted by ?, is a fundamental component in special relativity defined by ? = 1 / ?(1 - v²/c²), where v is the object's speed and c is the speed of light. It quantifies the relativistic effects such as time dilation, length contraction, and increases in energy and mass as an object's speed approaches the speed of light, serving as a bridge between classical and relativistic physics.

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