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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. A R P 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 osep 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 v 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.)

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. A R P 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 osep 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 v 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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There are three parts a, b and c please answer all
**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. The diagram shows: - A boat moving along a circular path from point A to B. - The circular curve has a radius \( R \), centered on point \( P \). - The dotted line represents the boat’s trajectory as it separates from the track at point B and enters free fall. - The pond is level with point \( P \). - Angle \( \phi \) marks the starting point, and angle \( \theta \) marks the impact angle. On this particular ride, the boat starts from rest at point A and slides down the slope without friction. **Tasks:** a) Determine the angle \( \phi_{\text{sep}} \) at which the boat will separate from the track. b) Determine the horizontal distance \( \Delta x \) (from point \( P \)) at which the boat strikes the pond surface. c) Determine the impact speed \( v_f \) and impact angle \( \theta \). **Hints:** - Derive a formula giving the maximum speed \( v_{\max} \) at which the boat can stay on the track, in terms of the angle \( \phi \). (Circular kinematics.) - Derive a formula for the speed \( v \) of the boat as it traverses the circular slope, in terms of the angle \( \phi \). (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. The diagram shows: - A boat moving along a circular path from point A to B. - The circular curve has a radius \( R \), centered on point \( P \). - The dotted line represents the boat’s trajectory as it separates from the track at point B and enters free fall. - The pond is level with point \( P \). - Angle \( \phi \) marks the starting point, and angle \( \theta \) marks the impact angle. On this particular ride, the boat starts from rest at point A and slides down the slope without friction. **Tasks:** a) Determine the angle \( \phi_{\text{sep}} \) at which the boat will separate from the track. b) Determine the horizontal distance \( \Delta x \) (from point \( P \)) at which the boat strikes the pond surface. c) Determine the impact speed \( v_f \) and impact angle \( \theta \). **Hints:** - Derive a formula giving the maximum speed \( v_{\max} \) at which the boat can stay on the track, in terms of the angle \( \phi \). (Circular kinematics.) - Derive a formula for the speed \( v \) of the boat as it traverses the circular slope, in terms of the angle \( \phi \). (Conservation of energy.)
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