# Unit of gradient/slope?

So I have a graph:

The value of the gradient/slope is $1.6±0.4$ and the value of the intercept is $0.9±0.4$. But what are the units of the graph? Is the unit of the gradient $v^2M^{-1}$? What about the unit of the intercept?

• The units of the gradient are the units of $y/x$ and the units of the $y$ intercept are the units of $y$. – John Rennie Aug 8 '14 at 10:12

## 3 Answers

Unit analysis starts starts with making sure the units match on both sides of your equation. It is important to remember that every variable in an equation has a value and a unit that goes with it. I usually keep track of units by using brackets. To solve for these pesky unknown units I'll use $[u_{i}]$ where $i$ is just a label for whatever variable we are looking at.

For your graph the fitting equation is linear. If we write it out in a way that tracks the units:

$$\begin{array}. y[u_y] = m[u_m] \cdot x[u_x] + b [u_b]\\ y[u_y] = m \cdot x [u_m] [u_x] + b [u_b] \end{array}$$

What need for the units on the left to match the units on the right, so that all of this extra algebra we introduced with the brackets cancels out. That means $[u_m] [u_x]=[u_y]$ and $[u_b]=[u_y]$ are the answers to your questions, once you plug in the units you do know.

From the axes labels you currently show, you would therefor write: $$\begin{array}. [u_y] = [u_m] [u_x]\\ \bigg[(\frac{m}{s})^2 \bigg]= [u_m] [g]\\ \bigg[\frac{m^2}{s^2 g} \bigg]= [u_m]\\ \end{array}$$ and for the intercept $$\begin{array}. [u_y] = [u_b]\\ \bigg[\frac{m^2}{s^2} \bigg]= [u_b]\\ \end{array}$$.

There is nothing like units of graph. The unit of slope (v^2)/M is meter^2/(second^2 gram)

Most physics exams would expect you to quote unit for the gradient.

However, the axes are labeled as quantity/unit so that the scale is a pure number. (unitless) By this token the gradient should also be a pure number.

If you wish to interpret the gradient as being some physical quantity, then you have to " put the units in" ie reverse the process by which the units were deliberately removed

## protected by Qmechanic♦May 12 '16 at 16:11

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