How to Change a Reflecting Tangent Line Into a Continuous Tangent Line

In this explainer, we will learn how to determine the instantaneous speed of an object by using a tangent to find the slope at a point on the object's distance–time graph.

Recall that speed is a measure of how fast an object is traveling. It is found by taking the magnitude of velocity or by dividing the distance an object moves over a given period of time. This means the units of speed are expressed in length/time, usually metres per second (m/s):

The graph below shows the distance in metres (-axis) over time in seconds (-axis) of an object that moves at a constant speed. Since the speed is constant, the line is straight.

From the smooth, unchanging line on this graph, we can determine this object's speed by calculating the slope of the line:

The slope of a line can be determined by taking the vertical difference between two points on the line and dividing it by the horizontal difference between those same two points:

When the slope of the line is a higher value, the speed of the object is likewise greater. Let's calculate the speed of the object at the end of the graph at the point shown in the graph below.

At the point on the graph, the object has moved a distance of 20 metres and has been moving for 4 seconds. Taking the difference for the vertical and horizontal, this gives

So, we find the speed is 5 m/s.

Not every object always travels at a consistent speed, however. Some distance-over-time lines look like the graph below.

In these cases, the speed must be calculated using a different method: using either instantaneous or average speed.

Instantaneous speed is the speed of an object at a specific moment in time (such as at ).

Average speed is the overall speed of an object over a given period of time (such as from to ).

Before going further, it must be said that when discussing distance versus displacement, this explainer focuses on distance because it is a magnitude. This is because instantaneous speed is used almost exclusively compared to instantaneous velocity: when looking at a singular point, we are looking for magnitude, not direction.

This means that when a slope is negative, we would still represent its speed as a positive value. The graph below is a displacement–time (not a distance–time) graph that shows two lines with the same magnitudes but different directions.

The slope of the red line is 1: and the slope of the blue line is :

If we are only concerned with the speed—which is a magnitude, not a vector that has a direction—we would just take the absolute value of the slope:

Firstly, let's look at how to find average speed. It is found the same way as regular speed by taking the distance and dividing it by the time taken but with the additional understanding that the speed may be changing as time goes on.

The graph below shows three distinct regions where the slope is different.

Measuring from to , we would find that the average speed is the same as the graph with a single straight line, since the start point and endpoint are the same:

When the line is going up or down, like from to , the speed is positive. When the line is flat, like from to , the speed is 0. If we find the slopes for these regions, we can then find the exact speeds for these time periods. Starting with the region from to ,

From to , the line is also straight. This means that the speed of the object is constant between these two times. We can calculate the speed in the same way as before:

As before, from to , the line is straight, meaning the speed is constant between these two times, so we can calculate the speed the same way again:

So, although the average speed of this object is 5 m/s, the actual speed of the object changes a lot during these 4 seconds.

Let's look at another example.

Example 1: Finding the Instantaneous Speed of a Person from a Displacement–Time Graph Consisting Only of Straight Lines

A boy moves along a straight line. On the graph, the blue line shows the displacement, , of the boy from his starting position over time, .

1. What is his speed 2 seconds after he starts walking?
2. What is his speed 6 seconds after he starts walking?

Answer

Part 1

This graph shows displacement, not distance, along its -axis. This means to get the speed, we will need to take the magnitude of the slope.

Let's find the slope of the line 2 seconds after the boy starts walking. Since the line is straight (meaning it has the same slope) from all the way to , we can choose the endpoint to be anywhere along the line. The formula for the slope of a line is so to find the slope 2 seconds after the boy starts walking, the vertical difference, displacement, is from 0 m to 4 m, and the horizontal displacement, time, is from to :

Taking the absolute value to make this a magnitude gives

After 2 seconds of walking, the boy is moving with a speed of 1 m/s.

Part 2

The boy's speed 6 seconds after he starts walking is calculated with the same method. Again, as long as the line is straight, it means the slope is constant. Because of this, we can choose any two points along this line, as long as they are on the same straight line segment.

For convenience, let's choose the start of the line at and our endpoint at : and taking the absolute value to get speed,

So, after 6 seconds of walking, the boy is moving with a speed of 0.25 m/s.

The method of measuring the start point and the endpoint only works when the lines on these graphs are straight. The graph below shows the height of a ball above the ground as it is thrown up in the air and falls back down again.

The slope of this curved line is changing too quickly to find the average speed, so it is now time to find the instantaneous speed—the speed at a particular point—using tangents.

Tangents are straight lines that just barely touch a curved line, such that both lines have the same exact slope at the point where they meet. The diagram below shows a dotted red tangent line intercepting a blue curved line.

Let's look at an example.

Example 2: Identifying Tangents to a Curve

An object moves along a straight line. On the graph, the blue line shows the displacement, , of the object from its starting position over time, .

Which of the three dashed lines is a tangent to the blue line at ?

1. The orange line
2. The red line
3. The purple line

Answer

When looking at the blue line, for every point in time , there must be a different displacement . The only time this would not be true would be when the line is flat, indicating that the object's displacement is not changing, but this does not occur on this curve.

The tangent line for this curve will pass exactly through a point with unique values of and . In this case, .

The orange line intersects with the blue line, but its slope does not quite match up. The orange line's slope is steeper, so it is not tangent to the blue line.

The red line's slope is much too small where it intersects with the blue line. It is not tangent.

The purple line matches the slope more closely than other lines where it touches the blue line, so C is the correct answer.

Touching the curved line at all is not sufficient to be called tangent. The lines need to just barely touch. To do this, draw just a tiny portion of the line at that point, then extend it as straight as you can outward. This is demonstrated in the diagram below.

Tangent lines touch exactly at a single point of the curve, with the combination of and values being unique.

Let's look at an example.

Example 3: Identifying Tangents to a Curve

An object moves along a straight line. On the graph, the blue line shows the displacement, , of the object from its starting position over time, .

1. Which of the three dashed lines is a tangent to the blue line at ?
   1. The red line
   2. The purple line
   3. The orange line
2. Which of the three dashed lines is a tangent to the blue line at ?
   1. The red line
   2. The purple line
   3. The orange line
3. Which of the three dashed lines is a tangent to the blue line at ?
   1. The purple line
   2. The orange line
   3. The red line

Answer

Part 1

When , we see that only one line is touching the curved blue line at that point: the purple line. No other line is near it, so the answer is B.

Part 2

When , the orange line is the only one touching the curved blue line. Out of all the lines, the orange one is the steepest, meaning it has the highest speed at that point. The correct answer is C.

Part 3

When , only the red line is touching the blue line at that point. Note also that the red line seemingly continues to touch the line as time progresses. This is because the blue line's slope nearly exactly matches the slope of the red line. The correct answer is C.

Now that we can see how tangent lines are drawn, we can begin to find instantaneous speed. Since speed is just the slope of a line, by finding the line at a point, we can find the slope at that point by subsequently measuring the line.

The examples below show how.

Example 4: Finding the Speed of an Object Given a Tangent to a Curve on a Displacement–Time Graph

A ball is thrown up in the air, and it falls back down to the ground. The height, , of the ball above the ground over time, , is shown on the graph by the blue line. The red line is a tangent to the blue line at . What is the speed of the ball at ?

Answer

The speed at the instant when can be found by determining the slope at that moment in time. To determine the slope, we need to have a straight line.

We are given the tangent line at . Because the line is straight, we can calculate the slope using the differences in and at the very ends of the line. We see that, for the tangent line, goes from 0 s to 3 s, and from 5 m to 20 m. Recall the equation for the slope of a straight line: so, substituting in the values gives

Height is a displacement, so we must take the absolute value to find the speed, which is a magnitude:

The ball's instantaneous speed at is 5 m/s.

Example 5: Finding the Instantaneous Speed of an Object Given a Displacement–Time Graph

A ball is thrown up in the air, and it falls back down to the ground. The height, , of the ball above the ground over time, , is shown on the graph by the blue line. What is the speed of the ball at ?

Answer

We are not given a tangent line for this problem, only told to find the speed of the ball at the instant of . We must first begin by creating a tangent line at that moment in time. We do this by taking a small portion of the slope at that point and continuing it along the whole graph.

This tangent line is, however, completely flat. It has no change in the -axis:

When the ball is at , its speed is 0 m/s. Remember that the steepness of a slope determines how fast an object is, so, when there is no slope, speed is 0.

Let's look more at what exactly a steeper slope looks like.

Example 6: Identifying the Instantaneous Speed of an Object at a Point on a Curve on a Displacement–Time Graph

The graph shows the displacement of a motorcycle in a race along a straight track. At which of the labeled points on the graph does the motorcycle have the greatest speed?

1. Point A
2. Point B
3. Point D
4. Point E
5. Point C

Answer

This is a displacement–time graph, not a distance–time graph, but since the slope at every marked point appears to be positive, there is no need to take the absolute value to make it positive.

We are looking for the point with the greatest speed at a point, the instantaneous speed. To find this instantaneous speed, we first must have a slope, and before we have a slope, we need to draw tangent lines for each point. These lines are shown in the diagram below.

Individually measuring the lines can be done to determine their slopes and thus the instantaneous speeds at these points, but we do not need to know exact numbers. We are just looking for the point with the greatest speed.

A steeper slope means a faster speed, so we just need to look for the steepest slope. The slope at point A is definitely the least steep slope, and steepest slope is that of the tangent line for point E. The steepest slope, and thus the point with the greatest speed, is the one at point E, or answer D.

Let's summarize what we have learned in this explainer.

Key Points

• Speed can be found on a distance–time graph by measuring the line's slope. If it is a displacement–time graph, the speed can be found by taking the absolute value of the line's slope.
• The tangent line can be created at a point by taking a small portion of the slope near the point and extending it straight outward.
• Instantaneous speed can be found at a point by creating a tangent line at that point and determining its slope.
• The sharper the slope of a line, the greater the speed.
• If the slope is flat, the speed is zero.

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Source: https://www.nagwa.com/en/explainers/906137309802/

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