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Chapter 5: Problem 20

We know that the velocity of an object with mass has an upper limit of \(c .\) Is there an upper limit on its momentum? Its energy? Explain.

Short Answer

Expert verified
In the context of special relativity, there is no upper limit on the momentum and energy of an object with mass, even though its velocity has an upper limit of "c". This is due to the relativistic momentum and energy formulae, which both approach infinity as the object's velocity approaches "c". Specifically, the momentum formula is \( p = \frac{m \cdot v}{\sqrt{1 - \frac{v^2}{c^2}}}\), and the energy formula is \( E = \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}}\).

Step by step solution

01

Recall the formula for momentum in special relativity

The momentum of an object with mass 'm' and velocity 'v' in the context of special relativity is given by the formula: \[ p = \frac{m \cdot v}{\sqrt{1 - \frac{v^2}{c^2}}}\]
02

Determine the upper limit of momentum

As 'v' approaches the speed of light 'c', the denominator of the fraction approaches zero. Therefore, when 'v' is close to 'c', the fraction approaches infinity, and so does the momentum 'p'. Thus, there is no upper limit on the momentum of an object with mass.
03

Recall the formula for energy in special relativity

The total energy of an object with mass 'm' and velocity 'v' in the context of special relativity is given by the formula: \[ E = \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}}\]
04

Determine the upper limit of energy

Similar to the momentum case, as 'v' approaches the speed of light 'c', the denominator of the fraction approaches zero. Therefore, when 'v' is close to 'c', the fraction approaches infinity, and so does the energy 'E'. Thus, there is no upper limit on the energy of an object with mass. In conclusion, even though the velocity of an object with mass has an upper limit of "c", there are no upper limits on its momentum and energy. This is because, as the object's velocity approaches "c", its momentum and energy both approach infinity due to the relativistic formulae.

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