Log Ride (object sliding down a circularly curved slope). In an amusement park ride, a boat moves slowly in a narrow channel of water. It then passes over a slope into a pool below as shown. The water in the channel ensures that there is very little friction. B Ax On this particular ride, the slope (the black arc through points A and B) is a circular curve of radius R, centered on point P. The dotted line shows the boat's trajectory. At some point B along the slope, the boat (and the water falling with it) will separate from the track and fall freely as shown. Note that the pond is level with point P. Considering the boat as a particle, assume it starts from rest at point A and slides down the slope without friction. a) Determine the angle psep at which the boat will separate from the track. b) Determine the horizontal distance Ax (from point P) at which the boat strikes the pond surface. c) Determine the impact speed vf and impact angle 0. Hints: Derive a formula giving the maximum speed vmax at which the boat can stay on the track, in terms of the angle p. (Circular kinematics.) Derive a formula for the speed v of the boat as it traverses the circular slope, in terms of the angle p. (Conservation of energy.)

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**Log Ride (object sliding down a circularly curved slope)**

In an amusement park ride, a boat moves slowly in a narrow channel of water. It then passes over a slope into a pool below as shown. The water in the channel ensures that there is very little friction.

<img src="log_ride_diagram.png" alt="Log Ride Diagram" />

**Diagram Explanation:**
The diagram shows a circular slope through points A and B with a radius \( R \), centered on point \( P \). The dotted line represents the trajectory of the boat. The boat separates from the track at point B and falls freely to the pond surface, level with point \( P \).

**Scenario Description:**
- The boat starts from rest at point A and slides down the slope without friction.
- The separation angle \( \phi_{\text{sep}} \) at which the boat separates from the track needs to be determined.
- Determine the horizontal distance \( \Delta x \) from point \( P \) at which the boat strikes the pond.
- Calculate the impact speed \( v_f \) and the impact angle \( \theta \).

**Hints:**
- Derive a formula for the maximum speed \( v_{\text{max}} \) at which the boat can stay on the track in terms of the angle \( \phi \). (Use circular kinematics.)
- Derive a formula for the speed \( v \) of the boat as it traverses the circular slope in terms of the angle \( \phi \). (Use conservation of energy.)
Transcribed Image Text:**Log Ride (object sliding down a circularly curved slope)** In an amusement park ride, a boat moves slowly in a narrow channel of water. It then passes over a slope into a pool below as shown. The water in the channel ensures that there is very little friction. <img src="log_ride_diagram.png" alt="Log Ride Diagram" /> **Diagram Explanation:** The diagram shows a circular slope through points A and B with a radius \( R \), centered on point \( P \). The dotted line represents the trajectory of the boat. The boat separates from the track at point B and falls freely to the pond surface, level with point \( P \). **Scenario Description:** - The boat starts from rest at point A and slides down the slope without friction. - The separation angle \( \phi_{\text{sep}} \) at which the boat separates from the track needs to be determined. - Determine the horizontal distance \( \Delta x \) from point \( P \) at which the boat strikes the pond. - Calculate the impact speed \( v_f \) and the impact angle \( \theta \). **Hints:** - Derive a formula for the maximum speed \( v_{\text{max}} \) at which the boat can stay on the track in terms of the angle \( \phi \). (Use circular kinematics.) - Derive a formula for the speed \( v \) of the boat as it traverses the circular slope in terms of the angle \( \phi \). (Use conservation of energy.)
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