### Example Problem: Calculating the Final Speed of a Roller Coaster **Scenario:** (a) **Question 1:** Determine the final speed of a roller coaster if it starts from rest at the top of a 20.0 m hill, assuming that the work done by frictional forces is negligible. (b) **Question 2:** Calculate the final speed of the roller coaster (assuming negligible friction) if its initial speed at the top of the hill is 5.00 m/s. **Key Principles:** - **Conservation of Energy:** The principle of conservation of mechanical energy states that the sum of potential energy (PE) and kinetic energy (KE) in a closed system remains constant, assuming no energy is lost to friction or air resistance. \[ PE + KE = \text{constant} \] \[ \Delta(PE + KE) = 0 \] **Diagram Explanation:** The problem likely includes a diagram of a roller coaster at the top of a hill with a height of 20.0 m. This setup illustrates the conversion of gravitational potential energy at the highest point of the hill to kinetic energy as the coaster descends, resulting in an increase in speed. Friction is assumed to be negligible, so the mechanical energy is conserved.
### Example Problem: Calculating the Final Speed of a Roller Coaster **Scenario:** (a) **Question 1:** Determine the final speed of a roller coaster if it starts from rest at the top of a 20.0 m hill, assuming that the work done by frictional forces is negligible. (b) **Question 2:** Calculate the final speed of the roller coaster (assuming negligible friction) if its initial speed at the top of the hill is 5.00 m/s. **Key Principles:** - **Conservation of Energy:** The principle of conservation of mechanical energy states that the sum of potential energy (PE) and kinetic energy (KE) in a closed system remains constant, assuming no energy is lost to friction or air resistance. \[ PE + KE = \text{constant} \] \[ \Delta(PE + KE) = 0 \] **Diagram Explanation:** The problem likely includes a diagram of a roller coaster at the top of a hill with a height of 20.0 m. This setup illustrates the conversion of gravitational potential energy at the highest point of the hill to kinetic energy as the coaster descends, resulting in an increase in speed. Friction is assumed to be negligible, so the mechanical energy is conserved.
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![### Example Problem: Calculating the Final Speed of a Roller Coaster
**Scenario:**
(a) **Question 1:** Determine the final speed of a roller coaster if it starts from rest at the top of a 20.0 m hill, assuming that the work done by frictional forces is negligible.
(b) **Question 2:** Calculate the final speed of the roller coaster (assuming negligible friction) if its initial speed at the top of the hill is 5.00 m/s.
**Key Principles:**
- **Conservation of Energy:** The principle of conservation of mechanical energy states that the sum of potential energy (PE) and kinetic energy (KE) in a closed system remains constant, assuming no energy is lost to friction or air resistance.
\[ PE + KE = \text{constant} \]
\[ \Delta(PE + KE) = 0 \]
**Diagram Explanation:**
The problem likely includes a diagram of a roller coaster at the top of a hill with a height of 20.0 m. This setup illustrates the conversion of gravitational potential energy at the highest point of the hill to kinetic energy as the coaster descends, resulting in an increase in speed. Friction is assumed to be negligible, so the mechanical energy is conserved.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fadefa1c0-cf1c-4809-9011-6430afec44fc%2F56aa039c-113c-4cf1-800e-88172596dfe3%2Fgrveeoc.jpeg&w=3840&q=75)
Transcribed Image Text:### Example Problem: Calculating the Final Speed of a Roller Coaster
**Scenario:**
(a) **Question 1:** Determine the final speed of a roller coaster if it starts from rest at the top of a 20.0 m hill, assuming that the work done by frictional forces is negligible.
(b) **Question 2:** Calculate the final speed of the roller coaster (assuming negligible friction) if its initial speed at the top of the hill is 5.00 m/s.
**Key Principles:**
- **Conservation of Energy:** The principle of conservation of mechanical energy states that the sum of potential energy (PE) and kinetic energy (KE) in a closed system remains constant, assuming no energy is lost to friction or air resistance.
\[ PE + KE = \text{constant} \]
\[ \Delta(PE + KE) = 0 \]
**Diagram Explanation:**
The problem likely includes a diagram of a roller coaster at the top of a hill with a height of 20.0 m. This setup illustrates the conversion of gravitational potential energy at the highest point of the hill to kinetic energy as the coaster descends, resulting in an increase in speed. Friction is assumed to be negligible, so the mechanical energy is conserved.
Expert Solution
Step 1
From the law of conservation of energy, the total energy in a conservative system always remains constant.
A conservative system is one in which there are no non-conservative forces, such as forces of friction, air resistance etc.
Total energy of any object is equal to the sum of its kinetic and potential energies, given as
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