Question Video: Identifying Acceleration from a Distance–Time Graph | Nagwa Question Video: Identifying Acceleration from a Distance–Time Graph | Nagwa

# Question Video: Identifying Acceleration from a Distance–Time Graph Science • Third Year of Preparatory School

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Which of the following most correctly describes the motion represented by the following distance–time graph? The time intervals shown are equal. [A] The solid and dashed lines represent the same uniformly accelerated motion. [B] The solid line represents uniform motion, and the dashed line represents accelerated motion. [C] The solid line represents accelerated motion, and the dashed line also represents accelerated motion but with a different value of acceleration. [D] The solid line represents accelerated motion, and the dashed line represents uniform motion.

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### Video Transcript

Which of the following most correctly describes the motion represented by the following distance–time graph? The time intervals shown are equal. Is it (A) the solid and dashed lines represent the same uniformly accelerated motion? Is it (B) the solid line represents uniform motion, and the dashed line represents accelerated motion? (C) The solid line represents accelerated motion, and the dashed line also represents accelerated motion but with a different value of acceleration. Or (D) the solid line represents accelerated motion, and the dashed line represents uniform motion.

The distance–time graph shows a line that has been drawn in two sections. One section of the line has been drawn as a solid line, and the other section of the line has been drawn as a dashed line.

Looking at the solid section of the line, we see that it curves smoothly. A line curving this way represents a gradual increase in speed of an object. For this section of the line, we can draw tangent lines that touch the line at different time values. We see that the slope of each tangent increases as the time value increases. This indicates that the speed of the object is increasing or, in other words, that the object is accelerating.

Now let’s look at the dashed part of the line. There are two time intervals defined on this part of the graph. These time intervals are shown by the diagram in the question to be equal. We can see though that the distances traveled in these two time intervals are not equal.

If we mark the start and end distances for each time interval as 𝑑 sub one and 𝑑 sub two for the first time interval, which starts at time 𝑡 sub one and ends at 𝑡 sub two, we can calculate the average speed for the first time interval. If we mark the start and end distances for the second time interval as 𝑑 sub three and 𝑑 sub four, which starts at time 𝑡 sub three and ends at 𝑡 sub four, we can calculate the average speed for the second time interval.

The first time interval shows a smaller distance traveled than the second interval, which also tells us that the object is traveling at a greater average speed during the second time interval than in the first time interval. In other words, the object increased in speed. We can see then that the object also accelerates in the section of the graph where the line is dashed. When we draw tangents to the dashed line, shown in green, the gradient of these tangents, representing the speed of the object, also increases as time increases, which demonstrates that the speed of the object is increasing in this part of the graph.

So the object is accelerating in both sections of the graph, but is the acceleration equal in both sections? The detail that tells us that the acceleration changes when the first section ends and the second section begins is that the line does not curve smoothly at the time that first section ends and the second section begins. The time that this occurs is shown as 𝑡.

We can see how the distance would need to have changed with time if the line had curved smoothly after 𝑡 in the same way that it did before 𝑡. And this line does not match the line in the question. We can see how the distance would need to have changed with the time if the line had curved smoothly before 𝑡 in the same way that it did after 𝑡. This line does not match the line in the question either.

So for the accelerations in the two sections to be equal, the line would have to curve smoothly in the same way before and after 𝑡. The line shown in the question does not do this, and so the acceleration must have changed at 𝑡.

We see then that the acceleration represented by the first section of the line does not equal the acceleration represented by the second section of the line. The correct answer is (C). The solid line represents accelerated motion, and the dashed line also represents accelerated motion but with a different value of acceleration.

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