To determine the percentage of initial mechanical energy that is dissipated by friction, we need to calculate the initial potential energy and the final kinetic energy of the roller coaster cars.
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Calculate the initial potential energy (PE_initial): The potential energy at the top of the drop can be calculated using the formula: \[ PE_{\text{initial}} = mgh \] where:
- \(m\) is the mass of the roller coaster cars (which will cancel out later),
- \(g\) is the acceleration due to gravity (\(9.81 , \text{m/s}^2\)),
- \(h\) is the height of the drop (68.6 m).
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Calculate the final kinetic energy (KE_final): The kinetic energy at the bottom of the drop can be calculated using the formula: \[ KE_{\text{final}} = \frac{1}{2} mv^2 \] where:
- \(v\) is the speed at the end of the drop (35.6 m/s).
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Initial potential energy (using mass \(m\)): \[ PE_{\text{initial}} = m \cdot 9.81 \cdot 68.6 \]
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Final kinetic energy (using mass \(m\)): \[ KE_{\text{final}} = \frac{1}{2} m (35.6)^2 \]
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Now we will simplify the expressions: The potential energy simplifies to: \[ PE_{\text{initial}} = m \cdot 9.81 \cdot 68.6 \approx 672.616 m \] The kinetic energy simplifies to: \[ KE_{\text{final}} = \frac{1}{2} m \cdot (35.6)^2 = \frac{1}{2} m \cdot 1267.36 \approx 633.68 m \]
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Determine how much mechanical energy is dissipated by friction: The initial mechanical energy is equal to the initial potential energy: \[ E_{\text{initial}} = PE_{\text{initial}} = 672.616 m \] The final mechanical energy is equal to the final kinetic energy: \[ E_{\text{final}} = KE_{\text{final}} = 633.68 m \]
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Calculate the energy dissipated by friction: \[ E_{\text{dissipated}} = E_{\text{initial}} - E_{\text{final}} = 672.616 m - 633.68 m = 38.936 m \]
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Now let's find the percentage of energy dissipated: \[ \text{Percentage dissipated} = \frac{E_{\text{dissipated}}}{E_{\text{initial}}} \times 100 = \frac{38.936 m}{672.616 m} \times 100 \]
The mass \(m\) cancels out: \[ \text{Percentage dissipated} = \frac{38.936}{672.616} \times 100 \approx 5.79% \]
Final result: Approximately 5.79% of the initial mechanical energy is dissipated by friction.