Interactive worksheet on the law of conservation of energy using roller coaster diagrams.

Let's solve this step-by-step based on the "All About Roller Coasters Interactive Activity" worksheet.

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🔹 Diagram 1: Matching Energy Percentages to Points

We are given a roller coaster track with labeled points: A, B, C, D, F. We must assign the correct energy percentages (PE = Potential Energy, KE = Kinetic Energy) to each point using the following options:

| Energy State |
|--------------|
| 0% PE, 100% KE |
| 35% PE, 65% KE – Gaining smaller amount of PE |
| 40% PE, 60% KE – Gaining KE |
| 80% PE, 20% KE – Gaining PE |
| 60% PE, 40% KE – Gaining PE |
| 100% PE, 0% KE |

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#### ✔ Understanding Energy Changes:
- At the highest point: Maximum potential energy (PE), minimum kinetic energy (KE) → 100% PE, 0% KE
- At the lowest point: Minimum PE, maximum KE → 0% PE, 100% KE
- As the coaster climbs up, it gains PE and loses KE
- As it descends, it gains KE and loses PE

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#### Let’s analyze each point:

Point A
- This is at the start of the climb, just after the initial drop or low point.
- It's going up → gaining PE
- But not yet at peak → so PE is less than 100%
- Still has speed → KE is high
- Looking at choices: 60% PE, 40% KE – Gaining PE fits well

✔ A → 60% PE, 40% KE, Gaining PE

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Point B
- This is the highest point on the track
- At the top → maximum height, so maximum PE
- Speed is minimal (momentarily zero at peak)
- So: 100% PE, 0% KE

✔ B → 100% PE, 0% KE

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Point C
- On the downhill slope from B
- The coaster is descending, so losing PE, gaining KE
- Not at bottom yet → some PE remains
- From the list: “40% PE, 60% KE, Gaining KE” matches this phase

✔ C → 40% PE, 60% KE, Gaining KE

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Point D
- At the lowest point of the first dip
- Lowest height → minimum PE
- Maximum speed → maximum KE
- So: 0% PE, 100% KE

✔ D → 0% PE, 100% KE

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Point F
- On a smaller hill after D
- Going up again → gaining PE
- But not as high as B → PE is less than 100%
- Still moving fast → KE is still significant
- The label says "Gaining smaller amount of PE"
- So: 35% PE, 65% KE – Gaining smaller amount of PE

✔ F → 35% PE, 65% KE – Gaining smaller amount of PE

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Now let's fill in the boxes:

```
[100% PE, 0% KE]
↑
B
/ \
/ \
A C
| |
[60% PE, 40% KE, Gaining PE] [40% PE, 60% KE, Gaining KE]
| |
↓ ↓
[0% PE, 100% KE] [35% PE, 65% KE, Gaining smaller amount of PE]
D F
```

Wait — the diagram shows only one box above B, one below D, etc. Let’s match positions:

From image layout:
- Box above B → B
- Box below D → D
- Box left of A → A
- Box right of C → C
- Box below F → F

So:

- Left of A: 60% PE, 40% KE, Gaining PE
- Above B: 100% PE, 0% KE
- Right of C: 40% PE, 60% KE, Gaining KE
- Below D: 0% PE, 100% KE
- Below F: 35% PE, 65% KE, Gaining smaller amount of PE

✔ Final Diagram 1 Answers:

| Point | Energy State |
|-------|--------------|
| A | 60% PE, 40% KE, Gaining PE |
| B | 100% PE, 0% KE |
| C | 40% PE, 60% KE, Gaining KE |
| D | 0% PE, 100% KE |
| F | 35% PE, 65% KE, Gaining smaller amount of PE |

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🔹 Diagram 2: Analyzing the Second Roller Coaster

This diagram shows a roller coaster with three hills:
- First hill: 5.0 m high (point A)
- Second hill: lower than first
- Third hill: marked with height h and question mark (?)

The coaster starts at A, goes down, up to second hill, then up to third hill.

We are asked:

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#### 1. What potential problem do you see with this roller coaster?

🔍 Observation:
The third hill (C) appears higher than the first hill (A), which is 5.0 m.
But the coaster starts at A, so its initial gravitational potential energy (PE) is based on that height.

If the third hill is higher than the first, the coaster will not have enough energy to reach the top because:

- Total mechanical energy = Initial PE + KE
- But if it starts at rest or low speed at A, all energy comes from height
- If h > 5.0 m, it cannot make it over unless there's an engine or boost

✘ Problem: The third hill is taller than the first hill, but the coaster cannot gain more energy than it started with (assuming no motors).

✔ Answer:
> The third hill (C) is higher than the initial hill (A), but the roller coaster does not have enough gravitational potential energy to reach the top of the third hill due to conservation of energy.

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#### 2. Explain what would happen as a result of this problem.

> The roller coaster would not have enough kinetic energy to climb the third hill. It would slow down rapidly, possibly stop before reaching the top, and then roll back down due to gravity. This could be dangerous for riders and could lead to a crash.

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#### 3. How would you fix this problem?

> Lower the height of the third hill so that it is equal to or lower than the first hill (5.0 m). Alternatively, add a boost mechanism (like a motor or launch system) between the second and third hills to give extra energy.

But since we're focusing on conservation of energy, the best fix is:

✔ Reduce the height of the third hill (h) so that it is ≤ 5.0 m.

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#### 4. Explain why your solution would work.

> Because of the law of conservation of energy, the total mechanical energy (PE + KE) remains constant (ignoring friction). The coaster starts with maximum PE at the first hill. As it moves, it converts PE to KE and back. But it cannot go higher than its starting point without additional energy input.

By making the third hill shorter than or equal to the first hill, the coaster can reach the top using only the energy it started with.

✔ Answer:
> Reducing the height of the third hill ensures the coaster has enough gravitational potential energy to reach the top, since it can only rise to a height equal to or less than its initial height without additional energy input.

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✔ Final Answers Summary

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Diagram 1: Energy Assignments

| Point | Energy Description |
|-------|--------------------|
| A | 60% PE, 40% KE, Gaining PE |
| B | 100% PE, 0% KE |
| C | 40% PE, 60% KE, Gaining KE |
| D | 0% PE, 100% KE |
| F | 35% PE, 65% KE, Gaining smaller amount of PE |

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Diagram 2: Questions

1. What potential problem do you see with this roller coaster?
> The third hill is taller than the first hill, but the roller coaster doesn’t have enough energy to reach the top due to conservation of energy.

2. Explain what would happen as a result of this problem.
> The roller coaster would slow down, fail to reach the top of the third hill, and roll back down, possibly causing a safety hazard.

3. How would you fix this problem?
> Lower the height of the third hill so it is equal to or less than the first hill.

4. Explain why your solution would work.
> According to the law of conservation of energy, the coaster cannot rise higher than its starting point without external energy input. By reducing the height of the third hill, it can safely reach the top using only the initial gravitational potential energy.

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✔ This demonstrates how potential and kinetic energy convert and how energy conservation limits the design of roller coasters.

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