Chapter 16, Problem 23A

What happens to the momentum of a massive object as its speed gets closer and closer to the speed of light?

To determine

To explain: The effect on the momentum of a massive object as the speed approaches closer to the speed of light.

Answer to Problem 23A

The momentum will approach to infinity.

Explanation of Solution

Introduction:

The relativistic mass formula gives the mass of an object moving with relativistic speed which can be written as,

  $m=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$$\left(1\right)$

Where, $m$ is relativistic mass,

  $m_0$ is the mass of an object at rest

  $v$ is the velocity of an object

  $c$ is the speed of light, $c=3\times {10}^8\text{m/s}$

Momentum $\left(p\right)$ of an object of mass $\left(m\right)$ moving with velocity $\left(v\right)$ is given by,

  $p=mv$

Substituting for $m$ from equation $\left(1\right)$ , the expression for momentum of an object moving with relativistic speed can be written as,

  $p=\frac{m_{0}v}{\sqrt{1-\frac{v^2}{c^2}}}$ $\left(2\right)$

The expression for relativistic momentum (equation $\left(2\right)$ ) clearly shows that as the velocity of an object increases its momentum also increases. As the relativistic speed of an object reaches closer and closer to the speed of light, the value of the denominator in the equation $\left(2\right)$ approaches to zero. And hence its momentum reaches close to infinity.

Conclusion:

As the speed of a massive object gets closer and closer to the speed of light, its momentum approaches to infinity.