Energy and Loop-the-Loop: Roller Coaster Dynamics

We model a frictionless roller coaster starting at height 60 m, descending into a vertical loop of radius 10 m, and then climbing another hill. The key physics principle is conservation of mechanical energy and, at the loop, the centripetal force condition to stay on the track.

---

1. Energy Conservation

Because the track is frictionless, the total mechanical energy (potential + kinetic) is conserved:

$$E = K + U = \frac{1}{2}mv^2 + mgh = \text{constant}.$$

Let:

• $h_0 = 60\,\text{m}$ be the starting height,
• $g = 9.8\,\text{m/s}^2$,
• $h$ be the height of the car at some point on the track,
• $v$ be its speed at that point.

Starting from rest, initial energy is:

$$E_0 = mgh_0.$$

At a general point:

$$\frac{1}{2}mv^2 + mgh = mgh_0$$

Cancel $m$:

$$\frac{1}{2}v^2 + gh = gh_0$$

Solve for $v$:

$$v = \sqrt{2g(h_0 - h)}.$$

This formula governs the speed everywhere along the track as long as we know the height $h$.

---

2. The Vertical Loop

The loop has radius $R = 10\,\text{m}$. We will parameterize the position in the loop by an angle $\theta$:

• $\theta = 0$ at the bottom of the loop,
• $\theta = \pi/2$ at the side,
• $\theta = \pi$ at the top of the loop.

Take the vertical coordinate of the bottom of the loop to be $h_\text{bottom}$. Then the height of the car at angle $\theta$ around the loop is:

$$h(\theta) = h_\text{bottom} + R(1 - \cos \theta).$$

Plug into energy conservation to get speed in the loop:

$$v(\theta) = \sqrt{2g\big(h_0 - h(\theta)\big)}.$$

In the visualization, we place the bottom of the loop significantly below the starting hill so that the coaster has gained considerable speed before entering the loop.

---

3. Condition to Stay on the Track at the Top of the Loop

At the top of the loop ($\theta = \pi$), the forces on the car (mass $m$) are:

• Weight: $mg$ downward,
• Normal force from the track: $N$ downward (toward center) when in contact.

The centripetal force requirement is:

$$\frac{mv^2}{R} = mg + N.$$

For the car to just maintain contact at the top, the smallest possible normal force is $N = 0$. Then:

$$\frac{mv_\text{top}^2}{R} = mg \Rightarrow v_\text{top}^2 = gR.$$

This gives the minimum speed at the top of the loop to avoid losing contact:

$$v_\text{top, min} = \sqrt{gR}.$$

Using energy conservation between the start and the top of the loop, we can find the minimum starting height required to reach this speed without falling off. But in this scenario, we already have $h_0 = 60\,\text{m}$, which is quite high relative to $R = 10\,\text{m}$, so the coaster will comfortably maintain contact.

In the visualization, we compute the actual speed at the top from energy and compare it to $\sqrt{gR}$. The difference gives us a sense of how "safe" the loop is.

---

4. Climbing the Second Hill

After exiting the loop, the roller coaster climbs another hill. Since energy is still conserved (no friction), the maximum height it can reach is the original height (if the path allows it):

$$\frac{1}{2}mv_\text{exit}^2 + mgh_\text{exit} = mgh_0.$$

If the second hill's peak is at height $h_2$, then the car can reach this top if

$$h_2 \le h_0.$$

The visualization lets you adjust the height of the second hill and instantly see whether the coaster can reach the top (it will slow down and potentially roll back if the hill is too high).

---

5. What the Visualization Shows

The canvas depicts:

1. A starting hill at 60 m.
2. A smooth track descending into a vertical loop of radius 10 m.
3. A second hill, whose height you can adjust.
4. A moving coaster car following the track, with speed determined by energy conservation.
5. A visual indicator of the centripetal condition at the top of the loop:
   • A colored arc or glow around the top of the loop that gets brighter when the speed safely exceeds $\sqrt{gR}$.

You can:

• Control the gravitational acceleration $g$ to explore how different planets affect the ride.
• Adjust the initial height and second hill height to see how energy limits the motion.
• Control the simulation speed and play/pause.

All of this is rendered with a Cartesian-style coordinate transform onto a dark slate background with neon colors.

Use the controls to experiment:

• Lowering the start height makes the car slower; watch what happens at the top of the loop.
• Raising $g$ (like going to a high-gravity planet) increases the needed speed to stay in contact at the top.
• Raising the second hill height shows how energy conservation limits how far the coaster can climb.