How To Find Slope Of Position Time Graph: Exhaustive Insights And Facts


The slope of the position and time graph gives the value of velocity. This article is about how to find slope of position time graph. 

The ratio of change in the x-axis to the change in the y-axis gives the value of the slope of a graph. For a position-time graph, we take time on the x-axis and therefore position on the y-axis. On evaluating the value of the slope, you can know the magnitude and the direction of the velocity. 

The visual representation and relation between the time and the position of a particle are depicted by the position-time graph. For calculating the slope, a general formula is used. 

What is the formula for the slope of a graph

The general formula used to find the slope of a graph is:

Slope=rise/run

[latex]Slope = \frac{rise}{run}[/latex]

Run is equal to the change in the horizontal axis, whereas rise equals the change in the vertical axis. Let us derive the standard formula for the slope of the graph. 

The change in the horizontal that is x-axis is calculated as:

[latex]\Delta x = x_{2} – x_{1}[/latex]

The change in the vertical that is the y-axis is calculated as;

[latex]\Delta y = y_{2} – y_{1}[/latex]

For any given position-time graph, first of all, plot the points and then join them. You will get the slope of the graph. The next thing is to take two points on the slope. Suppose we mark one point as A and the other as B. 

Now, look at the above figure. The coordinates of A are [latex](x_{1}, y_{1})[/latex] , and the coordinates of B are [latex](x_{2}, y_{2})[/latex] . As we know that slope is the ratio of run and rise that is;

[latex]Slope = \frac{rise}{run}[/latex]

Run is used for nothing but the change in the horizontal axis that is the x-axis. And the rise is the change in the vertical axis that is the y-axis. Therefore the slope formula becomes. 

Slope=(y2-y1)/(x2-x1)

[latex]slope = \frac{\Delta y}{\Delta x}[/latex]

[latex]slope = \frac{y_{2} – y_{1}}{x_{2} – x_{1}}[/latex]

We know that the y-axis denotes the position and x-axis time. The formula of velocity is;

v=ds/dt

[latex]v = \frac{\mathrm{d s} }{\mathrm{d} t}[/latex]

Here s is the displacement, and t is the time.

On comparing the above equations, it is clear that; 

slope = velocity

[latex]slope = velocity[/latex]

So the slope of the position time graph gives the value of velocity.

Table showing motion of a school bus

The above table shows the motion of a school bus, and we have to draw the graph and find the slope that is its velocity.  The very first step is to take time on the x-axis and position it on the y-axis. After that, start plotting the points. 

Once all the points are plotted, join them, and you’ll get the slope. To calculate the value of this slope, take any two points on the slope. And mark the coordinates. Suppose in the above graph we take points as A (20, 5) and B (10, 3). 

Substitute these values in the slope formula:

Slope=(y2-y1)/(x2-x1)

Slope=(5-3)/(20-10)

Slope=2/10

Slope=0.2

[latex]slope = \frac{y_{2} – y_{1}}{x_{2} – x_{1}}[/latex]

[latex]slope = \frac{5 – 3}{20 – 10}[/latex]

[latex]slope = \frac{2}{10}[/latex]

[latex]slope = 0.2 [/latex]

Now the next question that arises is how to find slope of the position time graph when the slope is parallel to the time axis. We mark the points A (10, 4 ) and B (25, 4 ). On substituting the values in the slope formula, we get:

Slope=(y2-y1)/(x2-x1)

Slope=(4-4)/(25-10)

Slope=0/10

Slope=0

[latex]slope = \frac{y_{2} – y_{1}}{x_{2} – x_{1}}[/latex]

[latex]slope = \frac{4 – 4}{25 – 10}[/latex]

[latex]slope = \frac{0}{10}[/latex]

[latex]slope = 0[/latex]

Therefore, the slope of the graph is zero, and hence the velocity of the object is zero. 

One important point to note is velocity is a vector quantity. That means that it must have a directional factor. When we graph the position and time of an object, the only way to understand that it is moving forward or backward is by its sign. The positive value of the slope indicates the forward velocity. If it has a negative sign, then the object is moving backward.

Positive and Negative Slope

Look at the graph shown above. To find the slope or velocity of it, the very first step is to mark the points. And then substitute them in the formula we get the value of slope along with the sign. Here in figure (i) the slope is positive and in figure (ii) it is negative, that is moving backward. Thus we can tell now that the slope can be positive, negative, or even zero. 

By calculating the slope of a given graph, we can know all about its motion, from its position at the exact time to the velocity. The slope is the ratio of position and time. We know that the unit of position is the meter and that of time is the second. Therefore: 

Slope=m.s-1

[latex]slope = m s^{-1}[/latex]

And we know that it is the unit of velocity. So this is also one way to know that the slope of position and time graph gives us the value of velocity. So this was all how to find the slope of a position time graph. For more information or any doubt, check the FAQs mentioned below or comment in the comment box. 

Frequently Asked Questions (FAQs)

Explain position vs. time graph. 

Plotting the time and position of an object graphically is the positon-time graph. 

The position time graph describes the movement of a particle. In this graph, the value of time is taken on the x-axis and positioned on the y-axisBy plotting the coordinates, we can find the slope of the graph.  

What is the slope of a graph?

The general definition of slope is the ratio of ‘rise’ and ‘run’ of the particular graph. 

The steepness of a line is its slope. By plotting the coordinates of the graph and then joining them, you get the slope. The positive, negative, and zero are the possible values of the slope of a graph. The value of slope gives the value some particular physical quantity. 

How to find slope of position time graph?

The slope of position time graph indicates the value of the velocity of the object. 

The slope is calculated by using the general formula that is:

[latex]slope = \frac{\Delta y}{\Delta x}[/latex]

Here,

[latex]\Delta x = x_{2} – x_{1}[/latex]

[latex]\Delta y= y_{2} – y_{1}[/latex]

so, the slope formula becomes:

Slope=(y2-y1)/(x2-x1)

[latex]slope = \frac{y_{2} – y_{1}}{x_{2} – x_{1}}[/latex]

What does the zero slope value infer? 

There can be a case when the value of the slope of the graph becomes zero.  

For the position time graph, the zero slope means that the line would be parallel to the time axis. It means that the position component won’t change even if the time changes. This simply infers that the object is at the position of rest; it is not moving at all. 

How to know the direction of velocity by the slope? 

The velocity is the vector quantity; it is associated with the direction. 

If the slope is in a downward direction, then its value would be negative, which means that the direction of velocity would be backward. If the slope rises up from going left to the right, then the velocity would be in the forward direction. 

Rabiya Khalid

Hi,  I am Rabiya Khalid, currently pursuing my masters in Mathematics. Article writing is my passion and I have been professionally writing for more than a year now. Being a science student, I have a knack for reading and writing about science and everything related to it. If you like what I write you can connect with me on LinkedIn: https://www.linkedin.com/mwlite/in/rabiya-khalid-bba02921a In my free time, I let out my creative side on a canvas. You can check my paintings at: https://www.instagram.com/chronicles_studio/

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