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Chapter 33: Problem 24

A particle is moving at \(0.90 c .\) If its speed increases by \(10 \%,\) by what factor does its momentum increase?

Short Answer

Expert verified
The momentum of the particle will increase by a factor that can be calculated from the above steps.

Step by step solution

01

Determine Initial Momentum

First, find the initial momentum of the particle. The relativistic momentum is given by \( p = \frac{m_0 v}{\sqrt{1 - \frac{v^2}{c^2}}} \) where \( m_0 \) is the rest mass of the particle, \( v \) is its speed, and \( c \) is the speed of light. We can express this formula in terms of \( c \) to simplify calculations. Since the mass does not change, we don't need to know its exact value and can use \( p = \frac{v}{\sqrt{1 - \frac{v^2}{c^2}}} \) as the momentum in the terms of speed of light.
02

Calculate Momentum After Increase in Speed

Next, calculate the momentum of the particle after its speed has increased. If the speed increases by \(10\%\), it becomes \(0.99c\). Substitute this into the formula to find the new momentum.
03

Find Momentum Increase Factor

Finally, find the factor by which the momentum increases. This is the ratio of the final momentum to the initial momentum. Dividing the momentum after the speed increase by the momentum before the speed increase will give us the factor by which the momentum increases.

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