Question Video: Comparing the Distances Moved by Two Objects Plotted on the Same Distance-Time Graph Physics

The red and blue lines show the change in distance moved with time for two objects that have the same speed as each other. The distance moved by the object represented by the red line has a value of 𝑑₁ and the distance moved by the object represented by the blue line has a value of 𝑑₂. The time between the start and end of the motion of the object represented by the red line is Δ𝑑₁ and the time between the start and end of the motion of the object represented by the blue line is Δ𝑑₂. Which of the following is the relationship between 𝑑₁ and 𝑑₂? [A] 𝑑₁ > 𝑑₂ [B] 𝑑₁ < 𝑑₂ [C] 𝑑₁ = 𝑑₂

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Video Transcript

The red and blue lines show the change in distance moved with time for two objects that have the same speed as each other. The distance moved by the object represented by the red line has a value of 𝑑 one and the distance moved by the object represented by the blue line has a value of 𝑑 two. The time between the start and end of the motion of the object represented by the red line is Δ𝑑 one and the time between the start and end of the motion of the object represented by the blue line is Δ𝑑 two. Which of the following is the relationship between 𝑑 one and 𝑑 two? Is it (a) 𝑑 one is greater than 𝑑 two, (b) 𝑑 one is less than 𝑑 two, or (c) 𝑑 one is equal to 𝑑 two?

If we take a look at our graph, we can see that we have distance on the vertical axis and time on the horizontal axis. For this part of the question, we’re interested in the distances 𝑑 one and 𝑑 two, where 𝑑 one is the distance moved by the object represented by the red line and 𝑑 two is the distance for the object represented by the blue line. The dashed lines on here show that the red and blue lines start at the same distance and also end at the same distance. The total distance moved by each object is equal to the change in value on the vertical axis only. So, the total distance moved by the two objects is the same, and therefore, 𝑑 one is equal to 𝑑 two.

Next, we need to find which of the following is the relationship between Δ𝑑 one and Δ𝑑 two. Is it (a) Δ𝑑 one is equal to Δ𝑑 two, (b) Δ𝑑 one is less than Δ𝑑 two, or (c) Δ𝑑 one is greater than Δ𝑑 two?

Now, remember that the question specifies that the values Δ𝑑 one and Δ𝑑 two refer to the time between the start and end of the motion of the objects. Therefore, Δ𝑑 one starts when the object represented by the red line starts moving, which is around here, and ends when the object stops moving, which is around here. So, it’s this interval of time here that’s represented by Δ𝑑 one, whereas the object represented by the blue line starts moving sometime later. So, Δ𝑑 two starts around here and ends here. So, this interval here is Δ𝑑 two.

Now, there are no grid markings on this graph, so it’s difficult to tell by eye if those two intervals are the same. So, it’s helpful to recall that speed equals distance over time. Now, we found in the previous part of the question that 𝑑 one is equal to 𝑑 two. So, the two distances are the same. And in the question, we’re told that the two objects have the same speed. Therefore, the speed of the two objects is equal. And for two objects that have equal speeds and have covered equal distances, that means the time intervals must also be equal. Therefore, the relationship between Δ𝑑 one and Δ𝑑 two is (a) Δ𝑑 one is equal to Δ𝑑 two.

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