Roller Coaster Dynamics Lab: Conservation of Energy
Investigate the **Conservation of Mechanical Energy** ($ME$). In ideal systems, the sum of Potential Energy ($PE$) and Kinetic Energy ($KE$) remains constant: $ME = PE + KE = \text{constant}$.
Mission Briefing: Key Equations & Formulas
â–¼- **Potential Energy ($PE$):** $$ PE = m \cdot g \cdot h $$
- **Kinetic Energy ($KE$):** $$ KE = \frac{1}{2} m \cdot v^2 $$
$$ PE_{\text{start}} + KE_{\text{start}} = PE_{\text{end}} + KE_{\text{end}} $$
Work done by non-conservative forces ($W_{nc}$, e.g., friction) equals the change in mechanical energy: $$ W_{nc} = \Delta ME = ME_{\text{end}} - ME_{\text{start}} $$
Experiment 1: Ideal System Verification (Frictionless)
Simulate a frictionless track. The cart starts from rest ($v_{start}=0$) at height $h_{start}$.
Experiment 2: Work-Energy Challenge (Non-Ideal System)
The cart starts at $h_1$, passes through a friction zone, and stops at $h_{end}$. Calculate $v_{end}$ or $W_{nc}$.
In the absence of non-conservative forces (like friction or air drag), energy freely converts between $PE$ and $KE$. A roller coaster reaches its maximum speed ($KE$) at its lowest point ($h=0$), where its $PE$ is minimum.
Real-World Energy Loss (11th Std)
In a real system, the initial total energy is always greater than the final total energy. The difference is the work done against friction, which is always negative (energy loss). $$ W_{\text{friction}} = ME_{\text{final}} - ME_{\text{initial}} $$
The rate at which mechanical energy is transferred or converted is called Power ($P$). The average power is $$ P_{\text{avg}} = \frac{W}{\Delta t} $$.
Work is zero when the force applied is perpendicular to the displacement (e.g., gravity doing work on a cart moving on a horizontal surface).
