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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 B. C 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

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 B. C 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

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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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. The water in the channel ensures minimal friction. On this ride, the slope (black arc through points \(A\) and \(B\)) is a circular curve of radius \(R\), centered on point \(P\). The dotted line indicates the boat’s trajectory. At some point \(B\) along the slope, the boat (and the water with it) separates from the track to fall freely. Note that the pond is level with point \(P\). Assume the boat is a particle, starting from rest at point \(A\) and sliding down the slope without friction. ### Problems: 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 \(P\)) where the boat strikes the pond surface.** c) **Determine the impact speed \(v_f\) and impact angle \(\theta\).** ### Hints: - Derive a formula for the maximum speed \(v_{\text{max}}\) for the boat to stay on the track, in terms of the angle \(\phi\). (Circular kinematics.) - Derive a formula for the speed \(v\) of the boat on the circular slope, in terms of the angle \(\phi\). (Conservation of energy.) ### Diagram Explanation: The diagram illustrates the boat’s motion over a circular slope. The slope is drawn with a solid arc from point \(A\) to point \(B\), showing the circular path of radius \(R\) centered at point \(P\). The boat’s trajectory is depicted with a dotted line as it separates from the curve at point \(B\) and continues through the air, eventually hitting the horizontal surface of the pond level with \(P\) at a distance \(\Delta x\).
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. The water in the channel ensures minimal friction. On this ride, the slope (black arc through points \(A\) and \(B\)) is a circular curve of radius \(R\), centered on point \(P\). The dotted line indicates the boat’s trajectory. At some point \(B\) along the slope, the boat (and the water with it) separates from the track to fall freely. Note that the pond is level with point \(P\). Assume the boat is a particle, starting from rest at point \(A\) and sliding down the slope without friction. ### Problems: 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 \(P\)) where the boat strikes the pond surface.** c) **Determine the impact speed \(v_f\) and impact angle \(\theta\).** ### Hints: - Derive a formula for the maximum speed \(v_{\text{max}}\) for the boat to stay on the track, in terms of the angle \(\phi\). (Circular kinematics.) - Derive a formula for the speed \(v\) of the boat on the circular slope, in terms of the angle \(\phi\). (Conservation of energy.) ### Diagram Explanation: The diagram illustrates the boat’s motion over a circular slope. The slope is drawn with a solid arc from point \(A\) to point \(B\), showing the circular path of radius \(R\) centered at point \(P\). The boat’s trajectory is depicted with a dotted line as it separates from the curve at point \(B\) and continues through the air, eventually hitting the horizontal surface of the pond level with \(P\) at a distance \(\Delta x\).
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