Worksheet for analyzing motion using position vs. time graphs and motion maps.

Let’s solve each problem step by step.

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Problem 1:

We are given a position vs. time graph and asked to:

a) Draw a motion map with one dot for each second (from t=0 to t=7).
b) Describe the motion in words.

First, let’s read the graph carefully.

The graph has:
- Time (seconds) on x-axis: from 0 to 8
- Position (meters) on y-axis: from 0 to 4

Key points from the graph:
- At t = 0 s → position = 0 m
- At t = 1 s → position = 2 m
- At t = 2 s → position = 4 m
- At t = 3 s → position = 4 m (stays same)
- At t = 4 s → position = 3 m
- At t = 5 s → position = 2 m
- At t = 6 s → position = 1 m
- At t = 7 s → position = 0 m

So we can list positions per second:

| Time (s) | Position (m) |
|----------|--------------|
| 0 | 0 |
| 1 | 2 |
| 2 | 4 |
| 3 | 4 |
| 4 | 3 |
| 5 | 2 |
| 6 | 1 |
| 7 | 0 |

Now, draw the motion map:

On the number line from 0 m to 5 m, place dots at the correct positions for each second.

At t=0: dot at 0 m
t=1: dot at 2 m
t=2: dot at 4 m
t=3: dot at 4 m (same as before — so two dots stacked or side-by-side at 4m? Usually, we put them slightly offset vertically to show it's a different time)
t=4: dot at 3 m
t=5: dot at 2 m
t=6: dot at 1 m
t=7: dot at 0 m

Also, add arrows between dots to show direction of motion.

From t=0 to t=2: moving right (position increasing) → arrow pointing right
t=2 to t=3: not moving → no arrow or just a dot
t=3 to t=7: moving left (position decreasing) → arrows pointing left

Now describe the motion in words:

“The object starts at 0 meters and moves quickly to the right, reaching 4 meters at 2 seconds. It stays still at 4 meters for 1 second (until t=3). Then it moves back to the left at a steady speed, returning to 0 meters at 7 seconds.”

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Problem 2:

We are given a motion map and asked to:

a) Draw a position vs. time graph.
b) Describe the motion in words.

Look at the motion map:

It shows dots along a number line from 0 m to 5 m, with arrows showing direction.

Let’s interpret the motion map step by step.

Assume each row represents a different time interval, but actually — since it says “positions have been recorded with one dot each second”, we need to count how many dots there are and their positions over time.

Looking at the diagram:

There are 7 dots total (count them):

Let’s label them in order from left to right as they appear in time? Wait — actually, the arrows tell us the direction of motion.

Better approach: trace the path.

Start from the leftmost dot at 0 m (first dot), then next dot is also at 0 m? No — look again.

Actually, the motion map is drawn with multiple rows. Let me reconstruct based on standard interpretation.

Typically, in such diagrams, you read top to bottom or left to right as time progresses. But here, the arrows indicate movement.

Let me list the positions in order of time:

Looking at the bottom row first (often earliest time):

Bottom row:
- Dot at 0 m → then arrow to right → dot at 2 m → arrow to right → dot at 4 m

Middle row:
- Dot at 0 m → arrow to left? Wait, no — middle row has three dots at 0 m? That doesn’t make sense.

Wait — perhaps the diagram is meant to be read as follows:

Each horizontal set of dots and arrows represents the motion during that second.

But actually, looking more carefully — this is likely a single timeline where dots are placed at positions at each second, and arrows show direction between them.

Let me count the dots and assign times.

There are 7 dots total. So time goes from t=0 to t=6.

Let’s go from left to right as time increases? Not necessarily — because some dots are above others.

Standard way: In motion maps, if dots are stacked vertically at same position, it means the object was there for multiple seconds.

But here, the dots are arranged in rows. Let me try to interpret:

Top row:
- Three dots at 4 m, 3 m, 2 m? With arrows pointing left? Actually, top row has dots at approximately 4m, 3m, 2m with left-pointing arrows.

Middle row:
- Dots at 0m, 0m, 0m? With left-pointing arrows? That seems odd.

Bottom row:
- Dots at 0m, 2m, 4m with right-pointing arrows.

This is confusing. Perhaps the diagram is meant to be read as three separate motions? No, the instruction says “the following motion map” implying one object.

Alternative interpretation: The motion map shows the object starting at 0m, moving right to 2m, then to 4m (bottom row), then staying at 4m for a while? But there are no dots at 4m in middle or top.

Wait — let’s count all dots:

I see:
- Bottom row: 3 dots (at 0, 2, 4) with right arrows
- Middle row: 3 dots (all at 0?) with left arrows? That doesn't make sense.
- Top row: 3 dots (at 4, 3, 2) with left arrows

That would be 9 dots, but probably not.

Perhaps it's a single sequence:

Start at 0m (bottom left), move right to 2m, then to 4m (so t=0,1,2)

Then from 4m, move left to 3m, then to 2m (t=3,4,5)

Then from 2m, move left to 0m? But there's a dot at 0m in middle row.

Another idea: maybe the vertical stacking indicates time spent at that position.

For example:

At t=0: position 0m (bottom dot)
t=1: position 2m (bottom row, next dot)
t=2: position 4m (bottom row, last dot)
t=3: position 4m (but no dot shown? Or perhaps the top row starts here)

I think I need to assume that the motion map is drawn with time progressing from left to right across the entire diagram, and vertical position doesn't matter — only horizontal position matters for location, and arrows show direction.

So let's list the dots in order from left to right:

Leftmost dot: at 0m (say t=0)
Next dot to the right: at 2m (t=1) — arrow from 0 to 2
Next: at 4m (t=2) — arrow from 2 to 4
Then, next dot is at 3m (top row, but horizontally between 2 and 4? Approximately at 3m) — arrow from 4 to 3? But the arrow is pointing left, so yes.
Then next dot at 2m (top row) — arrow from 3 to 2
Then next dot at 0m (middle row) — arrow from 2 to 0? But that would skip 1m.

This is messy. Perhaps the diagram has:

- From t=0 to t=2: moving right from 0 to 4m in steps of 2m per second? So t=0:0m, t=1:2m, t=2:4m
- Then from t=3 to t=5: moving left from 4m to 0m? But at t=3:4m, t=4:2m, t=5:0m? But in the diagram, after 4m, there's a dot at 3m, then 2m, then 0m.

Let's count the dots again. I think there are 7 dots:

1. At 0m (bottom left)
2. At 2m (bottom middle)
3. At 4m (bottom right)
4. At 3m (top right-ish)
5. At 2m (top middle)
6. At 0m (middle left)
7. And another at 0m? Or is that the same?

Perhaps it's:

t=0: 0m
t=1: 2m
t=2: 4m
t=3: 3m
t=4: 2m
t=5: 1m? But no dot at 1m.
t=6: 0m

But in the diagram, after 2m in top row, there's a dot at 0m in middle row, which might be t=5 or t=6.

To simplify, let's assume the motion is:

- Starts at 0m at t=0
- Moves to 2m at t=1
- Moves to 4m at t=2
- Stays at 4m? No, next dot is at 3m, so at t=3: 3m
- t=4: 2m
- t=5: 1m? But no dot at 1m — in the diagram, after 2m, there's a dot at 0m, so perhaps t=5: 0m

But that would mean from 2m to 0m in one second, which is possible.

So let's define:

| Time (s) | Position (m) |
|----------|--------------|
| 0 | 0 |
| 1 | 2 |
| 2 | 4 |
| 3 | 3 |
| 4 | 2 |
| 5 | 0 | // assuming jump from 2 to 0, or perhaps it's 1m then 0m, but only 6 dots? We have 7 dots listed earlier.

I think I made a mistake. Let me recount the dots in the image description.

Upon closer inspection of the user's image description, the motion map for problem 2 has:

- A number line from 0m to 5m.
- Below it, there are dots with arrows.
- Specifically, there are three rows of dots:
- Bottom row: three dots at 0m, 2m, 4m with right-pointing arrows between them.
- Middle row: three dots at 0m, 0m, 0m? With left-pointing arrows? That doesn't make sense.
- Top row: three dots at 4m, 3m, 2m with left-pointing arrows.

This suggests that the object might be doing multiple things, but that can't be.

Perhaps the motion map is for an object that:

- First moves right from 0 to 4m in 2 seconds (dots at 0,2,4 at t=0,1,2)
- Then moves left from 4m to 2m in 2 seconds (dots at 4,3,2 at t=3,4,5)
- Then moves left from 2m to 0m in 1 second (dot at 0 at t=6)

But in the middle row, there are dots at 0m with left arrows, which might be a mistake or misinterpretation.

Another possibility: the middle row dots at 0m are for when the object is at 0m at different times, but with arrows indicating it was moving left to get there.

Let's assume the following sequence based on common textbook problems:

The object:
- Starts at 0m at t=0
- Moves right to 2m at t=1
- Moves right to 4m at t=2
- Moves left to 3m at t=3
- Moves left to 2m at t=4
- Moves left to 1m at t=5? But no dot at 1m.
- Moves left to 0m at t=6

In the diagram, after the top row's 2m, there is a dot at 0m in the middle row, so perhaps t=5: 0m, skipping 1m.

To resolve this, I'll use the most logical interpretation: the motion map shows 7 dots corresponding to t=0 to t=6, with positions:

t=0: 0m
t=1: 2m
t=2: 4m
t=3: 3m
t=4: 2m
t=5: 1m // even if not explicitly shown, it must be there for constant velocity? But the problem doesn't say constant velocity for this part.
t=6: 0m

But in the diagram, there is no dot at 1m; instead, there is a dot at 0m in the middle row after the 2m in top row.

Perhaps the dot at 0m in middle row is at t=5, and the object moved from 2m to 0m in one second, so average velocity, but for position vs time graph, we plot the position at each second.

So let's define the positions as:

From the motion map, reading the dots in chronological order (left to right, top to bottom or however):

Assume the bottom row is first: t=0:0m, t=1:2m, t=2:4m

Then top row: t=3:4m? No, top row has dots at 4m,3m,2m — so perhaps t=3:4m, t=4:3m, t=5:2m

Then middle row: t=6:0m (since there's a dot at 0m with left arrow, but if it's at t=6, and previous was at 2m at t=5, then from 2m to 0m in one second.

So positions:

t=0: 0m
t=1: 2m
t=2: 4m
t=3: 4m // if the first dot in top row is at 4m, but in the diagram, the top row dots are at 4,3,2, so likely t=3:4m, t=4:3m, t=5:2m
t=6: 0m // from middle row dot at 0m

But that's only 7 points: t=0 to t=6.

So table:

| Time (s) | Position (m) |
|----------|--------------|
| 0 | 0 |
| 1 | 2 |
| 2 | 4 |
| 3 | 4 | // if the top row starts with a dot at 4m
| 4 | 3 |
| 5 | 2 |
| 6 | 0 | // from middle row

But in the middle row, there are three dots at 0m with left arrows, which might indicate that at t=6,7,8 it is at 0m, but that would be more than 7 dots.

I think for the sake of solving, I'll assume the motion map has 7 dots for t=0 to t=6 with positions:

- t=0: 0m
- t=1: 2m
- t=2: 4m
- t=3: 3m
- t=4: 2m
- t=5: 1m // inferred, as it's missing but necessary for smooth motion
- t=6: 0m

But since the diagram may not have a dot at 1m, perhaps it's:

- t=0: 0m
- t=1: 2m
- t=2: 4m
- t=3: 3m
- t=4: 2m
- t=5: 0m // jumping from 2m to 0m

This is ambiguous, but in many worksheets, they expect you to connect the dots as given.

Looking back at the user's image description, for problem 2, the motion map has:

"0 m [number line] 5 m"

Then below, "dots with arrows":

- One dot at 0m with right arrow to next dot at 2m
- Then to 4m
- Then from 4m, left arrow to 3m
- Then to 2m
- Then from 2m, left arrow to 0m (in middle row)

And there are additional dots at 0m in middle row with left arrows, which might be errors or for emphasis.

To simplify, I'll take the positions as:

t=0: 0m
t=1: 2m
t=2: 4m
t=3: 3m
t=4: 2m
t=5: 0m // assuming direct move from 2m to 0m

So for the position vs. time graph, plot these points:

(0,0), (1,2), (2,4), (3,3), (4,2), (5,0)

Then connect them with straight lines since it's particle model (constant velocity between points).

Describe the motion:

"The object starts at 0 meters and moves right to 2 meters in the first second, then to 4 meters in the next second. It then moves left to 3 meters at 3 seconds, to 2 meters at 4 seconds, and finally back to 0 meters at 5 seconds."

But this has non-constant velocity, which is fine.

For accuracy, let's stick to what's visible.

Final decision for Problem 2:

Positions from motion map (interpreting the dots in order):

- At t=0: position 0 m (first dot)
- t=1: position 2 m (second dot, with arrow from 0 to 2)
- t=2: position 4 m (third dot, with arrow from 2 to 4)
- t=3: position 3 m (fourth dot, with arrow from 4 to 3)
- t=4: position 2 m (fifth dot, with arrow from 3 to 2)
- t=5: position 0 m (sixth dot, with arrow from 2 to 0) — assuming the dot at 0m in middle row is at t=5
- And there might be a seventh dot at 0m at t=6, but let's say t=5 is the last.

In the diagram, there are three dots in middle row at 0m, which might mean it stays at 0m for three seconds, but that would be later.

To match the number of dots, let's say there are 6 dots for t=0 to t=5.

So for the graph, plot:

Time 0: pos 0
Time 1: pos 2
Time 2: pos 4
Time 3: pos 3
Time 4: pos 2
Time 5: pos 0

Then the position vs. time graph will have lines connecting these points.

Description: "The object moves right from 0m to 4m in 2 seconds, then moves left, passing through 3m and 2m, and reaches 0m at 5 seconds."

Now, for the final answer, since the user wants the solution, I'll provide the descriptions and the graphs as described.

But since I can't draw here, I'll describe how to draw them.

For Problem 1 motion map: place dots at positions: t0:0m, t1:2m, t2:4m, t3:4m, t4:3m, t5:2m, t6:1m, t7:0m — wait, earlier I had t7:0m, but in the graph, at t=7, position=0, and t=6:1m, etc.

From the graph in Problem 1:

At t=0:0, t=1:2, t=2:4, t=3:4, t=4:3, t=5:2, t=6:1, t=7:0

So 8 points from t=0 to t=7.

For motion map, on the number line, put dots at those positions for each second.

For Problem 2, with the motion map, I'll assume the positions are:

From the diagram, let's say the object is at:
- t=0: 0m
- t=1: 2m
- t=2: 4m
- t=3: 3m
- t=4: 2m
- t=5: 0m (as the last dot)

So for the position vs. time graph, plot those points.

Now, for the final answer, since the user may expect the written description and the method to draw the graphs, I'll box the descriptions.

But the instruction is to provide the final answer after reasoning.

So for Problem 1:

Motion map: Dots at 0m (t=0), 2m (t=1), 4m (t=2), 4m (t=3), 3m (t=4), 2m (t=5), 1m (t=6), 0m (t=7). Arrows: right from t0 to t2, no arrow at t3 (or stationary), left from t3 to t7.

Description: "The object starts at 0 m and moves right at a constant speed to 4 m in 2 seconds. It remains at 4 m for 1 second. Then it moves left at a constant speed back to 0 m in 4 seconds."

For Problem 2:

Position vs. time graph: Plot points (0,0), (1,2), (2,4), (3,3), (4,2), (5,0) and connect with straight lines.

Description: "The object starts at 0 m and moves right to 2 m in the first second, then to 4 m in the next second. It then moves left to 3 m at 3 seconds, to 2 m at 4 seconds, and to 0 m at 5 seconds."

To confirm, let's ensure consistency.

For Problem 1, the graph shows from t=2 to t=3, position constant at 4m, so stationary.

From t=3 to t=7, position decreases from 4m to 0m in 4 seconds, so velocity = (0-4)/(7-3) = -1 m/s, constant.

From t=0 to t=2, position from 0 to 4m in 2 seconds, velocity = 2 m/s.

So description is accurate.

For Problem 2, with the motion map, if we have dots at 0,2,4,3,2,0 at t=0 to 5, then the motion is as described.

So final answers:

Final Answer:

For Problem 1:
- Motion map: Place dots at positions 0 m (t=0), 2 m (t=1), 4 m (t=2), 4 m (t=3), 3 m (t=4), 2 m (t=5), 1 m (t=6), 0 m (t=7). Add right-pointing arrows between t=0 to t=2, no arrow at t=3 (stationary), and left-pointing arrows from t=3 to t=7.
- Description: "The object starts at 0 meters and moves right at a constant speed to 4 meters in 2 seconds. It stays at 4 meters for 1 second. Then it moves left at a constant speed back to 0 meters in 4 seconds."

For Problem 2:
- Position vs. time graph: Plot the points (0,0), (1,2), (2,4), (3,3), (4,2), (5,0) on the grid and connect them with straight line segments.
- Description: "The object starts at 0 meters and moves right to 2 meters in the first second, then to 4 meters in the next second. It then moves left to 3 meters at 3 seconds, to 2 meters at 4 seconds, and to 0 meters at 5 seconds."