Engage: Phenomenon - The Unstoppable Rollercoaster?

Imagine building a rollercoaster with a massive starting drop. The cart screams down the hill, gaining immense speed. You know that as the cart drops, gravitational potential energy ($\Delta E_g$) is converted into kinetic energy ($\Delta E_k$). But wait—if you just let the cart coast along a flat track after the drop, does it go on forever?

In reality, rollercoasters eventually slow down and stop even without brakes. Where does the energy go? If we assume a perfectly frictionless system, energy simply flows between the track height (gravitational) and cart speed (kinetic). But the real world is noisy and bumpy!

Discussion Questions:

1. What real-world factors cause a rollercoaster to lose its kinetic energy over time?
2. How can we use a mathematical model to figure out exactly how much energy is “lost” (transferred to the environment) if we know the initial and final speeds and heights of the cart?

Explore: Rollercoaster Energy Computational Model Investigation

Objective: Use the Rollercoaster Energy Computational Model to determine how changes in kinetic and gravitational potential energy relate to thermal energy loss and overall energy flow into or out of the system.

Estimated Time: 30 minutes Materials: Computer with internet access, Rollercoaster Energy Computational Model simulation

Instructions:

1. Open the Rollercoaster Energy Computational Model.
2. You will see three sliders:
   • Change in Kinetic Energy ($\Delta E_k$): Represents how much the kinetic energy of the cart changes.
   • Change in Gravitational Potential Energy ($\Delta E_g$): Represents how much the potential energy of the cart changes (based on its height).
   • Energy Flow in/out ($E_{flow}$): Represents energy added (e.g., a motorized chain lift) or removed (e.g., brakes) from the overall system.
3. Observe the equations at the top right: $\Delta E_k + \Delta E_g + \Delta E_t = E_{flow}$
4. Adjust the sliders to simulate the following scenarios and record the calculated change in thermal energy ($\Delta E_t$) in your data table.

Data Collection:

Explain: Sensemaking

Use the data you collected to answer the following questions:

1. In Scenario A, what was the value of $\Delta E_t$? Why does this make sense based on the definition of a “frictionless” system?
2. Look at Scenario B. Even though the cart lost 300 J of gravitational potential energy, it only gained 250 J of kinetic energy. According to the simulation, what happened to the remaining 50 J of energy?
3. The computational model uses the equation: $\Delta E_t = E_{flow} - (\Delta E_k + \Delta E_g)$. Explain what this algebraic expression means in your own words. How does it demonstrate the Law of Conservation of Energy?
4. Look at the “Model Assumptions & Limitations” quiz at the bottom of the simulation. What are the limitations of assuming all “lost” energy becomes thermal energy ($\Delta E_t$)? What other forms might energy take in a real rollercoaster?

Elaborate/Evaluate: Designing a Rollercoaster Section

Task: You are a rollercoaster engineer tasked with designing a new track section where a cart enters with a certain amount of energy, goes over a small hill, and then speeds up down a large drop.

1. Create the algebraic description:
   • Define the boundaries of your system (e.g., the cart, the track, and the Earth).
   • Set the initial energies: The cart starts at the top of the first small hill with $E_k = 50 \text{ J}$ and $E_g = 200 \text{ J}$.
   • The track has a built-in magnetic booster that adds energy to the system: $E_{flow} = +100 \text{ J}$.
   • The final state is at the bottom of the large drop ($E_g = 0 \text{ J}$). Due to friction and sound, the system generates $\Delta E_t = +40 \text{ J}$ of thermal/other energy during the ride.
2. Calculate: Using the principle of conservation of energy (Total Initial Energy + $E_{flow}$ = Total Final Energy), what will the final Kinetic Energy ($E_k$) of the cart be at the bottom of the drop? Show your algebraic steps.
3. Analyze limitations: Describe one assumption you made in your calculation that might make your prediction of the final kinetic energy slightly inaccurate in real life.

Teacher Notes

NGSS Alignment: HS-PS3-1 Create a computational model to calculate the change in the energy of one component in a system when the change in energy of the other component(s) and energy flows in and out of the system are known.

Evidence Statement Mapping:

• Representation (1.a.i-iv): Students identify the components modeled, system boundaries, and represent initial/final energies and energy flows algebraically in the Elaborate/Evaluate section.
• Computational Modeling (2.a-b): Students use the simulation (a simple computer program) based on the conservation of energy to calculate changes in thermal energy when changes in other components ($\Delta E_k, \Delta E_g$) and flows ($E_{flow}$) are known during the Explore section.
• Analysis (3.b): Students identify and describe the limitations of the computational model (e.g., that not all ‘lost’ energy is purely thermal, but also sound or track deformation) in the Explain section and further analyze assumptions in their custom model in the Evaluate section.

Task Dimensions:

• SEP: Using Mathematics and Computational Thinking
• DCI: PS3.A: Definitions of Energy; PS3.B: Conservation of Energy and Energy Transfer
• CCC: Systems and System Models