To analyze a distance-time graph, we look at the slope of the lines representing the distance traveled over time. The slope indicates speed — a steeper slope suggests a faster speed, while a flatter slope indicates a slower speed.
a. Fastest Speed Representation
The line that represents travel at the fastest speed is the one with the steepest slope. This is because speed is calculated as distance divided by time (speed = distance/time). If a line rises quickly (i.e., it increases distance rapidly for a given unit of time), then the object represented by this line is traveling quickly. For example, if one line rises at a 45-degree angle while another line rises at a 30-degree angle, the 45-degree line has a greater slope and thus indicates a greater speed.
Justification: You can quantify speed by calculating the slope of each line segment. For instance, if one line rises 10 units of distance in 2 units of time, its speed is 5 units/time. If another line rises 20 units in 4 units of time, its speed is 5 units/time as well, but if the first line rises 20 units in only 2 units of time, its speed would be 10 units/time. Therefore, in such an example, the line that rises more steeply demonstrates the fastest speed.
b. Line for an Object Not Moving
For an object that is not moving with respect to the chosen frame of reference, the line that should be added to the graph would be horizontal, remaining parallel to the time axis (x-axis) at the distance corresponding to that object's position. This indicates that as time progresses, the distance remains constant, reflecting a speed of zero.
In summary, a horizontal line signifies that the distance does not change over time, which is characteristic of an object at rest relative to the observer. For example, if the object is 5 units away from the origin, the horizontal line would be drawn at the distance of 5 units, stretching across the width of the graph to represent that no further distance was covered as time progresses.
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