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Chapter 2: Problem 70

A runner is at the position \(x=0 \mathrm{m}\) when time \(t=0\) s. One hundred meters away is the finish line. Every ten seconds, this runner runs half the remaining distance to the finish line. During each ten-second segment, the runner has a constant velocity. For the first forty seconds of the motion, construct (a) the position-time graph and \((b)\) the velocity-time graph.

Short Answer

Expert verified
The position-time graph is a curve; the velocity-time graph has stepwise decreasing values.

Step by step solution

01

Determine the Position at Each Time Interval

The runner starts at position \(x = 0\) meters and the finish line is at \(x = 100\) meters. Every 10 seconds, the runner covers half the remaining distance to the finish line. Calculate the position at \(t = 10\) s, \(t = 20\) s, \(t = 30\) s, and \(t = 40\) s: - At \(t = 10\) s: The runner covers half of the 100 meters, reaching \(x = 50\) meters. - At \(t = 20\) s: Half of the remaining 50 meters, reaching \(x = 75\) meters. - At \(t = 30\) s: Half of the remaining 25 meters, reaching \(x = 87.5\) meters. - At \(t = 40\) s: Half of the remaining 12.5 meters, reaching \(x = 93.75\) meters.
02

Plot the Position-Time Graph

Using the positions calculated in Step 1, plot the position-time graph. The x-axis represents time in seconds, and the y-axis represents the position in meters. Plot the points: - (0, 0) - (10, 50) - (20, 75) - (30, 87.5) - (40, 93.75) Connect these points to form a curve.
03

Calculate the Velocity for Each Interval

Velocity can be determined by the change in position over time for each interval: - Between \(t = 0\) s and \(t = 10\) s: Velocity is \((50-0)/(10-0) = 5\) m/s. - Between \(t = 10\) s and \(t = 20\) s: Velocity is \((75-50)/(20-10) = 2.5\) m/s. - Between \(t = 20\) s and \(t = 30\) s: Velocity is \((87.5-75)/(30-20) = 1.25\) m/s. - Between \(t = 30\) s and \(t = 40\) s: Velocity is \((93.75-87.5)/(40-30) = 0.625\) m/s.
04

Plot the Velocity-Time Graph

Using the velocities calculated in Step 3, plot the velocity-time graph. The x-axis represents time in seconds, and the y-axis represents velocity in meters per second. Plot the horizontal line segments: - From \(t = 0\) s to \(t = 10\) s at \(v = 5\) m/s. - From \(t = 10\) s to \(t = 20\) s at \(v = 2.5\) m/s. - From \(t = 20\) s to \(t = 30\) s at \(v = 1.25\) m/s. - From \(t = 30\) s to \(t = 40\) s at \(v = 0.625\) m/s.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

velocity-time graph
A velocity-time graph illustrates how an object's velocity changes over a certain time period. The x-axis shows time, while the y-axis displays velocity in meters per second (m/s). For the runner's scenario, the velocity at any given period was constant. This means each 10-second segment can be depicted as a flat line on the velocity-time graph.
  • From 0 to 10 seconds, the runner ran with a velocity of 5 m/s. The graph will have a horizontal line at 5 m/s.
  • Between 10 and 20 seconds, the velocity decreased to 2.5 m/s.
  • From 20 to 30 seconds, it further decreased to 1.25 m/s.
  • Lastly, from 30 to 40 seconds, the velocity was 0.625 m/s.
This gradual decrease in velocity for each subsequent interval creates a descending step pattern on the graph.
Understanding these visuals helps grasp the concept of how velocity changes with time, providing clear insights into the runner's behavior over the intervals.
constant velocity
Constant velocity means that an object is moving in a straight line at a steady speed. During each 10-second interval, the runner maintained a constant velocity, which means the speed didn't change for those specific periods. In terms of physics, constant velocity signifies both uniform speed and direction. In our runner's case, while the direction remained constant (towards the finish line), the actual speed changed between intervals.
  • For each 10-second interval, the runner's velocity was constant, albeit different from the previous interval.
  • Constant velocity implies that on a position-time graph, the slope of the line segment steadily represents the velocity.
Though the velocities were different in successive intervals, within each interval, the runner maintained steady speeds, reaching set positions consistently during each timeframe.
distance-time relationship
The distance-time relationship is crucial to understanding how an object traverses space over time. The position-time graph embodies this relationship, showing how the runner closed the gap to the finish line.
  • Initially, the runner was at 0 meters at time zero.
  • Each subsequent 10-second interval brought the runner closer, covering half the remaining distance each time.
  • For example, from 0-10 seconds, the runner went from 0 to 50 meters.
  • Then, from 10 to 20 seconds, the position reached 75 meters, and so on.
This method of traveling half the remaining distance results in a series of diminishing increments, which when plotted as points on a position-time graph, form a curve that suggests the runner's slowing progress.
Understanding this relationship between time and distance visualizes how the decreasing velocity affects the runner's journey to the finish line.

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Most popular questions from this chapter

A woman and her dog are out for a morning run to the river, which is located 4.0 \(\mathrm{km}\) away. The woman runs at 2.5 \(\mathrm{m} / \mathrm{s}\) in a straight line. The dog is unleashed and runs back and forth at 4.5 \(\mathrm{m} / \mathrm{s}\) between his owner and the river, until the woman reaches the river. What is the total distance run by the dog?

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In preparation for this problem, review Conceptual Example 7 From the top of a cliff, a person uses a slingshot to fire a pebble straight downward, which is the negative direction. The initial speed of the pebble is 9.0 \(\mathrm{m} / \mathrm{s} .\) (a) What is the acceleration (magnitude and direction) of the pebble during the downward motion? Is the pebble decelerating? Explain. (b) After 0.50 \(\mathrm{s}\) , how far beneath the cliff top is the pebble?

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