Question

You are working with a team that is designing a new roller coaster-type amusement park ride for a major theme park. You are present for the testing of the ride, in which an empty $250-\mathrm{kg}$ car is sent along the entire ride. Near the end of the ride, the car is at near rest at the top of a $110-\mathrm{m}$ tall track. It then enters a final section, rolling down an undulating hill to ground level. The total length of track for this final section from the top to the ground is $250 \mathrm{~m}$. For the first $230 \mathrm{~m},$ a constant friction force of $50.0 \mathrm{~N}$ acts from computer-controlled brakes. For the last $20 \mathrm{~m},$ which is horizontal at ground level, the computer increases the friction force to a value required for the speed to be reduced to zero just as the car arrives at the point on the track at which the passengers exit. (a) Determine the required constant friction force for the last $20 \mathrm{~m}$ for the empty test car. (b) Find the highest speed reached by the car during the final section of track length $250 \mathrm{~m}$. (c) You are asked by your team supervisor to determine the answers to parts (a) and (b) for a fully loaded car with an upper limit of $450 \mathrm{~kg}$ of passenger mass. Find these new values. (d) The required friction force in part (c) is well within design limits. The fastest speed, however, is well below that of current leading rides, so you would like to increase the maximum speed. You can't make the tower taller above ground, so you decide to include a feature where part of the track goes underground. Determine the depth to which the underground part of the ride must go to increase the maximum speed to $55.0 \mathrm{~m} / \mathrm{s}$. Assume the overall length of the first part of the track remains at $230 \mathrm{~m}$ and the length of track from the top to the lowest underground point is $150 \mathrm{~m}$. The same 50.0 - $\mathrm{N}$ friction force acts on the entire $230-\mathrm{m}$ section of track. (e) Is the construction in part (d) feasible?

Name: Problem 25, Chapter 8: Physics for Scientists and Engineers with Modern Physics
Uploaded: 2022-10-04T23:49:36-08:00
Duration: 8 min 32 s
Channel: Jacob Schulze

   You are working with a team that is designing a new roller coaster-type amusement park ride for a major theme park. You are present for the testing of the ride, in which an empty $250-\mathrm{kg}$ car is sent along the entire ride. Near the end of the ride, the car is at near rest at the top of a $110-\mathrm{m}$ tall track. It then enters a final section, rolling down an undulating hill to ground level. The total length of track for this final section from the top to the ground is $250 \mathrm{~m}$. For the first $230 \mathrm{~m},$ a constant friction force of $50.0 \mathrm{~N}$ acts from computer-controlled brakes. For the last $20 \mathrm{~m},$ which is horizontal at ground level, the computer increases the friction force to a value required for the speed to be reduced to zero just as the car arrives at the point on the track at which the passengers exit. (a) Determine the required constant friction force for the last $20 \mathrm{~m}$ for the empty test car.
(b) Find the highest speed reached by the car during the final section of track length $250 \mathrm{~m}$.
(c) You are asked by your team supervisor to determine the answers to parts (a) and (b) for a fully loaded car with an upper limit of $450 \mathrm{~kg}$ of passenger mass. Find these new values.
(d) The required friction force in part (c) is well within design limits. The fastest speed, however, is well below that of current leading rides, so you would like to increase the maximum speed. You can't make the tower taller above ground, so you decide to include a feature where part of the track goes underground. Determine the depth to which the underground part of the ride must go to increase the maximum speed to $55.0 \mathrm{~m} / \mathrm{s}$. Assume the overall length of the first part of the track remains at $230 \mathrm{~m}$ and the length of track from the top to the lowest underground point is $150 \mathrm{~m}$. The same 50.0 - $\mathrm{N}$ friction force acts on the entire $230-\mathrm{m}$ section of track. (e) Is the construction in part
(d) feasible?

Physics for Scientists and Engineers with Modern Physics

Raymond A. Serway,… 10th Edition

Chapter 8, Problem 25

Instant Answer

Solved by Expert Jacob Schulze

10/04/2022

Step 1

The initial energy is given by the potential energy at the top of the track, which is $mgh$, where $m$ is the mass of the car, $g$ is the acceleration due to gravity, and $h$ is the height of the track. This energy is then used to overcome the friction force over Show more…

Show all steps

Thanks for your feedback!

You are working with a team that is designing a new roller coaster-type amusement park ride for a major theme park. You are present for the testing of the ride, in which an empty $250-\mathrm{kg}$ car is sent along the entire ride. Near the end of the ride, the car is at near rest at the top of a $110-\mathrm{m}$ tall track. It then enters a final section, rolling down an undulating hill to ground level. The total length of track for this final section from the top to the ground is $250 \mathrm{~m}$. For the first $230 \mathrm{~m},$ a constant friction force of $50.0 \mathrm{~N}$ acts from computer-controlled brakes. For the last $20 \mathrm{~m},$ which is horizontal at ground level, the computer increases the friction force to a value required for the speed to be reduced to zero just as the car arrives at the point on the track at which the passengers exit. (a) Determine the required constant friction force for the last $20 \mathrm{~m}$ for the empty test car. (b) Find the highest speed reached by the car during the final section of track length $250 \mathrm{~m}$. (c) You are asked by your team supervisor to determine the answers to parts (a) and (b) for a fully loaded car with an upper limit of $450 \mathrm{~kg}$ of passenger mass. Find these new values. (d) The required friction force in part (c) is well within design limits. The fastest speed, however, is well below that of current leading rides, so you would like to increase the maximum speed. You can't make the tower taller above ground, so you decide to include a feature where part of the track goes underground. Determine the depth to which the underground part of the ride must go to increase the maximum speed to $55.0 \mathrm{~m} / \mathrm{s}$. Assume the overall length of the first part of the track remains at $230 \mathrm{~m}$ and the length of track from the top to the lowest underground point is $150 \mathrm{~m}$. The same 50.0 - $\mathrm{N}$ friction force acts on the entire $230-\mathrm{m}$ section of track. (e) Is the construction in part (d) feasible?

Play audio

Feedback

Jacob Schulze and 96 other subject Physics 101 Mechanics educators are ready to help you.

Ask a new question

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Conservation of Mechanical Energy with Non?Conservative Forces

This concept involves the idea that the total mechanical energy (potential plus kinetic energy) of a system is conserved when only conservative forces act. However, when non?conservative forces like friction are present, these forces do work that dissipates energy (usually as heat), meaning that the initial mechanical energy is reduced by the amount of work done by such forces.

Work?Energy Principle

The work?energy principle states that the net work done on an object is equal to the change in its kinetic energy. This principle is particularly useful in scenarios where energy is transformed from potential to kinetic energy, and where non?conservative forces (e.g., friction) remove energy from the system, affecting the motion.

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field. It is calculated as the product of the mass, the gravitational acceleration, and the height above a reference point. This energy is converted to kinetic energy when the object descends, a key process in the analysis of systems involving vertical motion.

Role of Friction as a Dissipative Force

Friction is a non?conservative force that opposes motion, doing work that dissipates mechanical energy into other forms, typically thermal energy. Understanding how friction affects the total energy of a system is crucial when analyzing the dynamics of any moving object, especially in engineered systems where controlling speed or energy is important.

Energy Transfer and Transformation in Dynamical Systems

This concept refers to how energy is transferred between different forms (such as potential, kinetic, and thermal energy) within a system, and how design parameters (like mass and track length) influence these transformations. It involves applying energy conservation, accounting for energy losses, and understanding how modifications to system components result in different performance outcomes.

Recommended Videos

Transcript

00:01 In this problem, we have a roller coaster, an empty car, which starts at a height of 110 meters, and it goes down with the friction force of 50 newtons for its first 230 meters of going down.

00:17 The length of track in total is 250 meters.

00:20 And in those last 20 meters, it will increase the friction force, such that it will come to a stop.

00:30 So what we need to do for the first part of the problem is just figure out what is this friction force.

00:37 So let's do that.

00:40 Now, to do that, let's first calculate how much energy left over do we have after the first 230 meters.

00:50 So our starting out energy, we know, is simply going to be mgh.

00:56 And that is going to be equal to some kinetic energy.

01:01 We get out of these first 230 meters plus some amount of work and that work is going to be 50 newtons times 230 right since that is going to be the force times the distance do note that we don't have a potential energy term on the right and that's just due to the fact that we know that the last 20 meters is horizontal thus isn't going to be changing in height okay, so what this allows us to do is fine what the kinetic energy is going to be.

01:42 Since that will end up being our starting out energy for the final part.

01:48 So one half mv squared is going to be equal to mgh minus 50, 9 newtons times 230.

02:01 Okay, so this is the energy that's going to start out our last 20 meters, essentially.

02:08 And so what this is going to have to be equal to is the work of the friction, right? and so this is going to be equal to this force of friction that we don't know times the distance, which is 20 meters, right? and the reason this has to be equal to because this is to be the energy left over from just falling down those 110 meters.

02:37 And so essentially we have to get rid of all of our kinetic energy.

02:40 That's what we're doing here.

02:42 So let's solve for fk.

02:45 Fk is going to be equal to mghh minus 50 newtons times 230 meters divided by 20 meters.

03:01 Now what you get from this is 1 .29 times 10 to the 4 newtons.

03:10 So that is the friction force that we require in order to stop the car, quite a bit larger than the first 230 meters.

03:21 Okay, next part asks us to find the maximum velocity that the car experiences.

03:28 Well, we know that the maximum velocity is going to occur at the end of these 230 meters, since it has gone to full 110 meters in height downwards.

03:39 And so all it really comes down to is solving this for the velocity.

03:45 So all i have to do is solve that equation for velocity.

03:48 So let's just write that out.

03:51 So the velocity here, we can have to take a square root of everything on the right here.

04:02 230 meters, and i'll be dividing that by m, and also having to multiply everything on the top here by 2.

04:11 Okay, and that's square right of everything here.

04:14 So that's the velocity.

04:16 What you get, what you plug in the numbers is 45 .4 meters per second.

04:24 Okay, now the next part is, is to basically just do parts a and b with people in the car.

04:31 And so all we have to do for that part, ultimately, is to use what we found out, right? and that is that the friction force is equal to this, and that the velocity is equal to this...

Ace is your personal tutor. It breaks down any question with clear steps so you can learn.

Start Using Ace

Continue solving this problem

Create a free account to:

View full step-by-step solution
Ask follow-up questions with Ace AI
Save progress and study later

Please give Ace some feedback