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1. A roller coaster car of mass M is launched by a compressed spring of spring constant k. Assume that there is no friction or air drag. Starting from rest, the coaster is pushed by the compressed spring, climbs up and then down a hill of height 3.5R, and then goes around the inside of a loop of radius R. spring ME hill 3.5R Answer the questions in terms of any of the given quantities (M, R, g, k) as needed. a. What is the minimum distance the spring must be compressed for the coaster to make it over the hill? loop b. In the case of the coaster just making it over the first hill, what will be the normal force of the track on the coaster when it reaches the top of the loop? (Use conservation of mechanical energy to determine the speed at the top of the loop and the free body diagram method to determine the normal force.) C. When the spring is compressed a distance Xm, the coaster just barely gets over a hill of height H. How much would you need to compress the spring for the coaster car to make it over a hill of height 2H. Answer in terms of Xm-

1. A roller coaster car of mass M is launched by a compressed spring of spring constant k. Assume that there is no friction or air drag. Starting from rest, the coaster is pushed by the compressed spring, climbs up and then down a hill of height 3.5R, and then goes around the inside of a loop of radius R. spring ME hill 3.5R Answer the questions in terms of any of the given quantities (M, R, g, k) as needed. a. What is the minimum distance the spring must be compressed for the coaster to make it over the hill? loop b. In the case of the coaster just making it over the first hill, what will be the normal force of the track on the coaster when it reaches the top of the loop? (Use conservation of mechanical energy to determine the speed at the top of the loop and the free body diagram method to determine the normal force.) C. When the spring is compressed a distance Xm, the coaster just barely gets over a hill of height H. How much would you need to compress the spring for the coaster car to make it over a hill of height 2H. Answer in terms of Xm-

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### Activity 3.6 – Energy and Newton’s Laws Problems #### Problem Overview: 1. **Roller Coaster Dynamics** - A roller coaster car of mass \( M \) is launched by a compressed spring of spring constant \( k \). Assume that there is no friction or air drag. Starting from rest, the coaster is pushed by the compressed spring, climbs up and then down a hill of height \( 3.5R \), and then goes around the inside of a loop of radius \( R \). - **Diagram Description:** - The diagram shows a roller coaster car initially compressed by a spring. The car then climbs a hill of height \( 3.5R \), descends from the hill, and finally goes through a loop with radius \( R \). #### Questions: a. **Calculating Minimum Spring Compression:** - **Question:** What is the minimum distance the spring must be compressed for the coaster to make it over the hill? - **Variables:** \( M, R, g, k \) b. **Normal Force at the Top of the Loop:** - **Question:** In the case of the coaster just making it over the first hill, what will be the normal force of the track on the coaster when it reaches the top of the loop? - **Hint:** Use conservation of mechanical energy to determine the speed at the top of the loop and the free body diagram method to determine the normal force. c. **Spring Compression for Different Hill Heights:** - **Question:** When the spring is compressed a distance \( x_m \), the coaster just barely gets over a hill of height \( H \). How much would you need to compress the spring for the coaster car to make it over a hill of height \( 2H \)? - **Hint:** Answer in terms of \( x_m \). #### Concepts Involved: - **Conservation of Energy:** The total mechanical energy (kinetic + potential) of the coaster remains constant if there’s no friction. - **Newton’s Laws:** Particularly the relationship between force, mass, and acceleration, which can help determine the normal force in various parts of the coaster's journey. #### Explanation of the Schematic Diagram: - **Diagram Elements:** - **Spring:** Indicates initial compression. - **Hill:** Shows the height the roller coaster needs to climb (\(3.5R\)). - **
Transcribed Image Text:### Activity 3.6 – Energy and Newton’s Laws Problems #### Problem Overview: 1. **Roller Coaster Dynamics** - A roller coaster car of mass \( M \) is launched by a compressed spring of spring constant \( k \). Assume that there is no friction or air drag. Starting from rest, the coaster is pushed by the compressed spring, climbs up and then down a hill of height \( 3.5R \), and then goes around the inside of a loop of radius \( R \). - **Diagram Description:** - The diagram shows a roller coaster car initially compressed by a spring. The car then climbs a hill of height \( 3.5R \), descends from the hill, and finally goes through a loop with radius \( R \). #### Questions: a. **Calculating Minimum Spring Compression:** - **Question:** What is the minimum distance the spring must be compressed for the coaster to make it over the hill? - **Variables:** \( M, R, g, k \) b. **Normal Force at the Top of the Loop:** - **Question:** In the case of the coaster just making it over the first hill, what will be the normal force of the track on the coaster when it reaches the top of the loop? - **Hint:** Use conservation of mechanical energy to determine the speed at the top of the loop and the free body diagram method to determine the normal force. c. **Spring Compression for Different Hill Heights:** - **Question:** When the spring is compressed a distance \( x_m \), the coaster just barely gets over a hill of height \( H \). How much would you need to compress the spring for the coaster car to make it over a hill of height \( 2H \)? - **Hint:** Answer in terms of \( x_m \). #### Concepts Involved: - **Conservation of Energy:** The total mechanical energy (kinetic + potential) of the coaster remains constant if there’s no friction. - **Newton’s Laws:** Particularly the relationship between force, mass, and acceleration, which can help determine the normal force in various parts of the coaster's journey. #### Explanation of the Schematic Diagram: - **Diagram Elements:** - **Spring:** Indicates initial compression. - **Hill:** Shows the height the roller coaster needs to climb (\(3.5R\)). - **
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