Physics worksheet diagram showing a roller coaster track with points 1 through 5 to analyze energy transformation.

Let's solve this step-by-step using the Law of Conservation of Energy, which states:

> Total mechanical energy (Potential Energy + Kinetic Energy) remains constant in a closed system with no non-conservative forces (like friction or air resistance).

We assume no friction or air resistance in this problem, so mechanical energy is conserved.

---

Given:

- The roller coaster starts from rest at point 1 → initial kinetic energy = 0.
- It moves down the incline and goes through various points: 1, 2, 3, 4, and 5.
- We are to analyze potential energy (PE) and kinetic energy (KE) at these points.

---

a. Potential Energy (PE)

#### 🔹 What determines potential energy?
Gravitational potential energy is given by:
\[
PE = mgh
\]
where:
- \( m \) = mass (constant),
- \( g \) = acceleration due to gravity (constant),
- \( h \) = height above a reference point (ground level).

So, height determines PE — higher the point, greater the potential energy.

---

#### 🔹 Let’s compare heights at each point:

From the image:
- Point 1: Highest point → maximum height → highest PE
- Point 2: Lower than 1, but still elevated → medium-high PE
- Point 3: Loop-the-loop; top of loop is lower than point 1, but not as low as 4 or 5 → moderate PE
- Point 4: At the bottom of the loop → lowest height → lowest PE
- Point 5: Small hill, lower than 1, 2, and 3 → low PE, but higher than 4

Now let’s order them from most to least potential energy based on height:

1. Point 1 – highest
2. Point 2 – next highest
3. Point 3 – top of loop (lower than 2)
4. Point 5 – small hill (lower than 3)
5. Point 4 – lowest (bottom of loop)

✔ So, order from most to least potential energy:
\[
\boxed{1 > 2 > 3 > 5 > 4}
\]

➡️ What did you use to determine the order?
I used the height of each point relative to the ground. Since PE depends directly on height (\( PE = mgh \)), the taller the point, the greater the potential energy.

---

b. Kinetic Energy (KE)

#### 🔹 Law of Conservation of Energy:
\[
PE + KE = \text{Constant (Total Mechanical Energy)}
\]

Since total energy is conserved:
- When PE is high, KE must be low
- When PE is low, KE must be high

At point 1, the roller coaster starts from rest → KE = 0, all energy is potential.

As it moves down, PE decreases, KE increases.

---

#### 🔹 Now use the PE order from part (a) to deduce KE order:

| Point | PE Rank | KE Rank |
|-------|--------|--------|
| 1 | Highest | Lowest (since KE = 0 initially) |
| 2 | High | Low |
| 3 | Medium | Medium |
| 5 | Low | High |
| 4 | Lowest | Highest |

So, KE is greatest where PE is least, and vice versa.

Therefore, order from most to least kinetic energy:
\[
\boxed{4 > 5 > 3 > 2 > 1}
\]

✔ Explanation:
- At point 4, the roller coaster is at the lowest height → minimum PE → maximum KE
- At point 1, it's at rest → KE = 0 → minimum KE
- Points in between have intermediate values

➡️ What did you use to determine the order?
I used the conservation of mechanical energy and the inverse relationship between PE and KE. Since total energy is constant, when PE decreases, KE increases, and vice versa.

---

✔ Final Answers:

#### a. Potential Energy (from most to least):
\[
\boxed{1 > 2 > 3 > 5 > 4}
\]
Reason: Potential energy depends on height. Higher elevation = greater PE.

#### b. Kinetic Energy (from most to least):
\[
\boxed{4 > 5 > 3 > 2 > 1}
\]
Reason: By conservation of energy, when potential energy is low, kinetic energy is high. Point 4 has the lowest PE, so highest KE; point 1 has zero KE initially.

---

Let me know if you'd like a diagram or visual explanation!