Question

As the velocity of an object increases, so does its energy. What does this imply? a) At very low velocities, the object's energy is equal to its mass times \(c^{2}\). b) In order for an object with mass \(m\) to reach the speed of light, infinite energy is required. c) Only objects with \(m=0\) can travel at the speed of light. d) all of the above e) none of the above

Short answer

a) At very low velocities, the object's energy is equal to its mass times \(c^{2}\). b) In order for an object with mass \(m\) to reach the speed of light, infinite energy is required. c) Only objects with \(m=0\) can travel at the speed of light. d) All of the above. Answer: d) All of the above.

Step-by-step solution

Review the Equation for Relativistic Kinetic Energy

The relativistic kinetic energy equation is given by \(K = (\gamma - 1)mc^2\), where \(K\) represents the kinetic energy of an object, \(m\) is its mass, \(c\) is the speed of light, and \(\gamma\) is the Lorentz factor (\(\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\)). We will now analyze each statement given in the exercise based on this equation.

Analyze Statement a

Statement a) At very low velocities, the object's energy is equal to its mass times \(c^{2}\). This is incorrect. According to the equation, as the velocity (\(v\)) approaches 0, the Lorentz factor (\(\gamma\)) approaches 1, making the kinetic energy (\(K\)) approach 0. The energy would not be equal to its mass times \(c^2\) in this case.

Analyze Statement b

Statement b) In order for an object with mass \(m\) to reach the speed of light, infinite energy is required. As the object's velocity approaches the speed of light, the denominator of the Lorentz factor approaches 0, causing the factor itself to tend towards infinity. Consequently, the required kinetic energy \(K\) goes towards infinity. This statement is correct.

Analyze Statement c

Statement c) Only objects with \(m=0\) can travel at the speed of light. With a zero mass, the energy equation does not apply and the object can travel at the speed of light. Photons, for example, have no mass and travel at the speed of light. This statement is correct as well.

Choose the Correct Option

Since statements b and c are correct, while statement a is incorrect, the correct answer is d) all of the above, as all given options are in some way accurate.