/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 29 The cars of a roller-coaster rid... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 29

The cars of a roller-coaster ride have a speed of \(30 \mathrm{km} / \mathrm{h}\) as they pass over the top of the circular track. Neglect any friction and calculate their speed \(v\) when they reach the horizontal bottom position. At the top position, the radius of the circular path of their mass centers is \(18 \mathrm{m},\) and all six cars have the same mass.

Short Answer

Expert verified
The speed of the cars at the bottom is approximately 20.55 m/s.

Step by step solution

01

Understand the problem

We need to find the speed of the roller-coaster cars at the bottom of the circular track, given their speed at the top and the radius of the track. The conversion between the potential energy at the top and the kinetic energy at the bottom will help us solve this.
02

Identify key physics concepts

We will use the conservation of mechanical energy. At the top, the cars have both potential and kinetic energy. At the bottom, all potential energy is converted into kinetic energy. Neglecting friction means no energy is lost.
03

Write the conservation of energy equation

The total energy at the top is the sum of potential energy and kinetic energy, given by: \[ \frac{1}{2} mv_t^2 + mgh = \frac{1}{2} mv_b^2 \] Where \( m \) is mass, \( v_t \) is speed at the top, \( h \) is the height (which equals the radius here), and \( v_b \) is speed at the bottom.
04

Simplify the equation

Since mass \( m \) appears in all terms, it can be canceled out. The equation simplifies to: \[ \frac{1}{2} v_t^2 + gh = \frac{1}{2} v_b^2 \]
05

Calculate the potential and kinetic energy terms

Using \( v_t = 30 \text{ km/h} = 8.33 \text{ m/s} \), \( g = 9.8 \text{ m/s}^2 \), \( h = 18 \text{ m} \): - Kinetic energy term at the top: \( \frac{1}{2} (8.33)^2 \approx 34.73 \text{ m}^2/\text{s}^2 \)- Potential energy term: \( 9.8 \times 18 = 176.4 \text{ m}^2/\text{s}^2 \)
06

Plug into the simplified equation

Insert these values into the simplified conservation equation: \[ 34.73 + 176.4 = \frac{1}{2} v_b^2 \] \[ v_b^2 = 2 \times (34.73 + 176.4) = 422.26 \]
07

Solve for the final speed at the bottom

Take the square root of both sides to solve for \( v_b \):\[ v_b = \sqrt{422.26} \approx 20.55 \text{ m/s} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mechanical Energy
Mechanical energy is a vital concept in physics because it combines both kinetic and potential energy, providing a comprehensive view of an object's energy. It is the energy associated with the motion and position of an object and is highly significant in systems like roller coasters. In the case of the roller coaster problem, the mechanical energy at the top of the track is a combination of kinetic (
  • Energy due to the speed of the cars
) and potential energy (
  • Energy due to their height above the ground
).
By the law of conservation of energy, the total mechanical energy in a system remains constant if no external forces, such as friction, act on it. Therefore, when the cars go from the top to the bottom of the track, all potential energy is converted to kinetic energy. This idea allows us to predict the speed at the bottom based on the known conditions at the top, maintaining the same total mechanical energy throughout the ride.
Kinetic Energy
Kinetic energy is the energy an object possesses due to its motion. The faster an object moves, the more kinetic energy it has. For example, when the roller coaster cars are at the top of the track, they already have some kinetic energy, as indicated by their speed of 30 km/h (or 8.33 m/s).
To calculate kinetic energy, we use the formula: \[ KE = \frac{1}{2} mv^2 \] Here,
  • \( m \) is the mass of the object, and
  • \( v \) is its velocity.
At the top of the track, the cars have this velocity and thus possess kinetic energy. As they descend, this kinetic energy increases because the potential energy is being converted into kinetic, making the cars go faster. This increase continues until they reach the bottom, reflecting the maximum conversion of potential into kinetic energy as calculated in the exercise.
Potential Energy
Potential energy is the stored energy in an object due to its position. For our roller coaster example, the potential energy is greatest when the cars are at the top of the circular track because of their height. It is given by the formula:\[ PE = mgh \]Where:
  • \( m \) is mass,
  • \( g \) is the acceleration due to gravity (9.8 m/s虏), and
  • \( h \) is the height from the ground.
In this problem, the height equals the radius of the circular path, which is 18 m. When the cars are at the top, they have maximum potential energy because of this height difference. As they descend the track, this potential energy converts into kinetic energy, increasing the speed of the cars. At the bottom of the track, the potential energy is at its minimum (practically zero if the bottom is considered the reference height). Understanding this transition from potential to kinetic energy is crucial to explaining the change in speed as the roller coaster moves from top to bottom.

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Most popular questions from this chapter

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