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. А В 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, t boat (and the water falling with it) will separate from the track and fall freely as shown. Note that th 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 surfa 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.)

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)...
icon
Related questions
icon
Concept explainers
Topic Video
Question

I got Part B wrong and wasn't sure where at. Could someone walk me through it?

**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 presents a boat on a circular curve, showing its trajectory from point \( A \) along a circular path through points \( B \) to \( C \). The slope (black arc) is a circular curve of radius \( R \), centered on point \( P \). At point \( B \), the boat, along with the water, separates from the slope and falls freely into the pool. The pool is level with point \( P \).

Assuming the boat as a particle, it starts from rest at point \( A \) and slides down the slope without friction.

### Objectives:

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 \( v_{\text{max}} \) 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.)

**Graph Explanation:**

- **Arc \( AB \):** Shows the circular path of the boat down the slope.
- **Point \( P \):** The center of the circular curve with reference lines to the boat's trajectory.
- **Angle \( \phi \):** The angle between the radius at points \( A \) and \( B \).
- **Angle \( \theta \):** The angle of impact when the boat hits the water surface.
- **Line \( \Delta x \):** The horizontal distance from point \( P \) to where the boat hits the pond at point \( C \).

The diagram helps to visualize the boat's motion, effects of gravity, and separation from the slope, assisting in solving the physics-based questions.
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 presents a boat on a circular curve, showing its trajectory from point \( A \) along a circular path through points \( B \) to \( C \). The slope (black arc) is a circular curve of radius \( R \), centered on point \( P \). At point \( B \), the boat, along with the water, separates from the slope and falls freely into the pool. The pool is level with point \( P \). Assuming the boat as a particle, it starts from rest at point \( A \) and slides down the slope without friction. ### Objectives: 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 \( v_{\text{max}} \) 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.) **Graph Explanation:** - **Arc \( AB \):** Shows the circular path of the boat down the slope. - **Point \( P \):** The center of the circular curve with reference lines to the boat's trajectory. - **Angle \( \phi \):** The angle between the radius at points \( A \) and \( B \). - **Angle \( \theta \):** The angle of impact when the boat hits the water surface. - **Line \( \Delta x \):** The horizontal distance from point \( P \) to where the boat hits the pond at point \( C \). The diagram helps to visualize the boat's motion, effects of gravity, and separation from the slope, assisting in solving the physics-based questions.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 5 steps with 7 images

Blurred answer
Knowledge Booster
Displacement, velocity and acceleration
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.
Similar questions
  • SEE MORE QUESTIONS
Recommended textbooks for you
College Physics
College Physics
Physics
ISBN:
9781305952300
Author:
Raymond A. Serway, Chris Vuille
Publisher:
Cengage Learning
University Physics (14th Edition)
University Physics (14th Edition)
Physics
ISBN:
9780133969290
Author:
Hugh D. Young, Roger A. Freedman
Publisher:
PEARSON
Introduction To Quantum Mechanics
Introduction To Quantum Mechanics
Physics
ISBN:
9781107189638
Author:
Griffiths, David J., Schroeter, Darrell F.
Publisher:
Cambridge University Press
Physics for Scientists and Engineers
Physics for Scientists and Engineers
Physics
ISBN:
9781337553278
Author:
Raymond A. Serway, John W. Jewett
Publisher:
Cengage Learning
Lecture- Tutorials for Introductory Astronomy
Lecture- Tutorials for Introductory Astronomy
Physics
ISBN:
9780321820464
Author:
Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina Brissenden
Publisher:
Addison-Wesley
College Physics: A Strategic Approach (4th Editio…
College Physics: A Strategic Approach (4th Editio…
Physics
ISBN:
9780134609034
Author:
Randall D. Knight (Professor Emeritus), Brian Jones, Stuart Field
Publisher:
PEARSON