Interpreting Graphs Answer Key: Complete with ease | airSlate SignNow - Free Printable
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Step-by-step solution for: Interpreting Graphs Answer Key: Complete with ease | airSlate SignNow
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Step-by-step solution for: Interpreting Graphs Answer Key: Complete with ease | airSlate SignNow
Here’s the complete solution to Worksheet: Interpreting Graphs — Chapter 4: Linear Motion, based on the provided graph and questions.
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The graph shows distance (in meters) on the vertical axis vs. time (in seconds) on the horizontal axis for three runners: Alice, Bob, and Charlie.
- Alice’s line: Starts at (0,0), rises steeply, then curves upward — indicating increasing speed (acceleration).
- Bob’s line: Starts at (0,0), straight diagonal line — constant speed (linear motion).
- Charlie’s line: Starts at (0,0), rises slowly at first, then curves upward more sharply — also accelerating, but starts slower than Alice.
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## Answers & Explanations
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> Answer: The slope of a distance vs. time graph represents the speed of the runner. A steeper slope means faster speed; a shallower slope means slower speed. A curved line indicates changing speed (acceleration or deceleration).
> Explanation: Slope = rise/run = Δdistance / Δtime = speed. Since this is a distance-time graph, slope directly equals instantaneous speed. Straight lines = constant speed. Curved lines = changing speed.
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> Answer: Alice completed the 100m race in the least amount of time — approximately 10 seconds.
> Explanation: Look at where each runner’s line reaches 100 meters on the y-axis:
> - Alice reaches 100m at ~10 seconds.
> - Bob reaches 100m at ~15 seconds.
> - Charlie reaches 100m at ~18 seconds.
> So Alice is fastest.
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> Answer: Bob started out the fastest.
> Explanation: At the very beginning (t=0 to t≈2 seconds), Bob’s line has the steepest slope — meaning he covered more distance in less time initially compared to Alice and Charlie. Alice and Charlie start slower but accelerate later.
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> Answer: Charlie was accelerating — his speed was increasing.
> Explanation: Between 8 and 10.5 seconds, Charlie’s line becomes noticeably steeper. This means he was covering more distance per unit time — i.e., speeding up (accelerating). His curve bends upward, which is characteristic of acceleration on a distance-time graph.
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> Answer:
> - A straight line tells you the runner is moving at a constant speed (no acceleration).
> - A curved line tells you the runner is accelerating or decelerating (changing speed).
> Explanation: On a distance-time graph:
> - Straight line → constant slope → constant speed.
> - Curved line → changing slope → changing speed → acceleration.
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> Answer: Bob overtook Charlie at approximately ~50 meters and ~7.5 seconds.
> Explanation: Find where Bob’s and Charlie’s lines intersect. That point is where they are at the same distance at the same time — Bob catches up to Charlie. Visually, that intersection occurs around 50m and 7.5s.
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> Answer: The total displacement for each runner is 100 meters.
> Explanation: Displacement (Δd) is the change in position from start to finish. All three runners start at 0m and end at 100m. So:
> Δd = final position – initial position = 100m – 0m = 100m
> *Note:* Even though their paths differ (some accelerated, some didn’t), displacement only cares about start and end points — not the path taken.
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## ✔ Final Summary Table
| Question | Answer |
|----------|--------|
| 1. Slope meaning | Speed (steeper = faster) |
| 2. Fastest runner (100m) | Alice (~10s) |
| 3. Started fastest | Bob |
| 4. Charlie (8–10.5s) | Accelerating |
| 5. Straight line | Constant speed<br>Curved line → Changing speed |
| 6. Bob overtakes Charlie | ~50m at ~7.5s |
| 7. Total displacement | 100m for all |
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Let me know if you’d like a hand-drawn sketch or further clarification!
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Graph Overview (Graph 1: Distance vs. Time for Three Runners)
The graph shows distance (in meters) on the vertical axis vs. time (in seconds) on the horizontal axis for three runners: Alice, Bob, and Charlie.
- Alice’s line: Starts at (0,0), rises steeply, then curves upward — indicating increasing speed (acceleration).
- Bob’s line: Starts at (0,0), straight diagonal line — constant speed (linear motion).
- Charlie’s line: Starts at (0,0), rises slowly at first, then curves upward more sharply — also accelerating, but starts slower than Alice.
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## Answers & Explanations
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1. What does the slope of each line on the graph tell you?
> Answer: The slope of a distance vs. time graph represents the speed of the runner. A steeper slope means faster speed; a shallower slope means slower speed. A curved line indicates changing speed (acceleration or deceleration).
> Explanation: Slope = rise/run = Δdistance / Δtime = speed. Since this is a distance-time graph, slope directly equals instantaneous speed. Straight lines = constant speed. Curved lines = changing speed.
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2. Which runner completed the 100m race in the least amount of time? What was his time?
> Answer: Alice completed the 100m race in the least amount of time — approximately 10 seconds.
> Explanation: Look at where each runner’s line reaches 100 meters on the y-axis:
> - Alice reaches 100m at ~10 seconds.
> - Bob reaches 100m at ~15 seconds.
> - Charlie reaches 100m at ~18 seconds.
> So Alice is fastest.
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3. Which runner started out the fastest?
> Answer: Bob started out the fastest.
> Explanation: At the very beginning (t=0 to t≈2 seconds), Bob’s line has the steepest slope — meaning he covered more distance in less time initially compared to Alice and Charlie. Alice and Charlie start slower but accelerate later.
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4. What was Charlie doing between 8 seconds and 10.5 seconds?
> Answer: Charlie was accelerating — his speed was increasing.
> Explanation: Between 8 and 10.5 seconds, Charlie’s line becomes noticeably steeper. This means he was covering more distance per unit time — i.e., speeding up (accelerating). His curve bends upward, which is characteristic of acceleration on a distance-time graph.
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5. What does a straight line on this graph tell you? A curved line?
> Answer:
> - A straight line tells you the runner is moving at a constant speed (no acceleration).
> - A curved line tells you the runner is accelerating or decelerating (changing speed).
> Explanation: On a distance-time graph:
> - Straight line → constant slope → constant speed.
> - Curved line → changing slope → changing speed → acceleration.
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6. At what distance and time did Bob overtake Charlie?
> Answer: Bob overtook Charlie at approximately ~50 meters and ~7.5 seconds.
> Explanation: Find where Bob’s and Charlie’s lines intersect. That point is where they are at the same distance at the same time — Bob catches up to Charlie. Visually, that intersection occurs around 50m and 7.5s.
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7. What was the total displacement of all three runners? (Δd?)
> Answer: The total displacement for each runner is 100 meters.
> Explanation: Displacement (Δd) is the change in position from start to finish. All three runners start at 0m and end at 100m. So:
> Δd = final position – initial position = 100m – 0m = 100m
> *Note:* Even though their paths differ (some accelerated, some didn’t), displacement only cares about start and end points — not the path taken.
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## ✔ Final Summary Table
| Question | Answer |
|----------|--------|
| 1. Slope meaning | Speed (steeper = faster) |
| 2. Fastest runner (100m) | Alice (~10s) |
| 3. Started fastest | Bob |
| 4. Charlie (8–10.5s) | Accelerating |
| 5. Straight line | Constant speed<br>Curved line → Changing speed |
| 6. Bob overtakes Charlie | ~50m at ~7.5s |
| 7. Total displacement | 100m for all |
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Let me know if you’d like a hand-drawn sketch or further clarification!
Parent Tip: Review the logic above to help your child master the concept of interpreting graphs of functions worksheet.
