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Chapter 1: Problem 38

The relationship between kinetic energy \(K\) (SI unit $\left.\mathrm{kg} \cdot \mathrm{m}^{2} \cdot \mathrm{s}^{-2}\right)\( and momentum \)p\( is \)K=p^{2} /(2 m),\( where \)m$ stands for mass. What is the SI unit of momentum?

Short Answer

Expert verified
Answer: The SI unit of momentum is kg路m路s鈦宦�.

Step by step solution

01

Write the given formula

We have the provided formula relating kinetic energy (\(K\)) and momentum (\(p\)) as follows: $$K = \frac{p^2}{2m}$$
02

Identify the SI units of known variables

The SI unit of kinetic energy \(K\) is \(\mathrm{kg}\cdot\mathrm{m}^2\cdot\mathrm{s}^{-2}\), and the SI unit of mass \(m\) is \(\mathrm{kg}\). We need to find the SI unit of momentum \(p\).
03

Rewrite the formula in terms of SI units

Using the SI units for kinetic energy and mass, we can write the formula as follows: $$[\mathrm{kg}\cdot\mathrm{m}^2\cdot\mathrm{s}^{-2}] = \frac{[p]^2}{2[\mathrm{kg}]}$$
04

Solve for the SI unit of momentum

Now, we need to isolate [p], representing the SI unit of momentum. To do this, multiply both sides of the equation by \(2[\mathrm{kg}]\): $$2[\mathrm{kg}]\cdot[\mathrm{kg}\cdot\mathrm{m}^2\cdot\mathrm{s}^{-2}] = [p]^2$$ This simplifies to: $$[\mathrm{kg}^2\cdot\mathrm{m}^2\cdot\mathrm{s}^{-2}] = [p]^2$$ To find [p], we take the square root of both sides of the equation: $$\sqrt{[\mathrm{kg}^2\cdot\mathrm{m}^2\cdot\mathrm{s}^{-2}]} = [p]$$ Which gives us the SI unit of momentum: $$[\mathrm{kg}\cdot\mathrm{m}\cdot\mathrm{s}^{-1}] = [p]$$
05

State the final answer

The SI unit of momentum \(p\) is \(\mathrm{kg}\cdot\mathrm{m}\cdot\mathrm{s}^{-1}\).

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