Chapter 6: Problem 44
Calculate At an amusement park, a swimmer uses a water slide to enter the main pool. If the swimmer starts at rest, slides without friction, and descends through a vertical height of \(2.31 \mathrm{~m}\), what is her speed at the bottom of the slide?
Short Answer
Expert verified
The swimmer's speed at the bottom is 6.73 m/s.
Step by step solution
01
Identify Known Variables
We begin by identifying the key variables from the problem. The initial velocity (\( v_i \)) of the swimmer is 0 m/s (since the swimmer starts at rest), and the vertical height (\( h \)) that the swimmer descends is 2.31 m. We also know the acceleration due to gravity (\( g \)) is approximately 9.81 m/sĀ².
02
Select the Appropriate Formula
Since the swimmer is sliding without friction and starts from rest, we use the conservation of energy principle. The potential energy at the top of the slide is converted entirely into kinetic energy at the bottom. The formula for this is: \[ mgh = \frac{1}{2} mv^2 \]where \( m \) is mass, \( g \) is gravitational acceleration, \( h \) is height, and \( v \) is final velocity.
03
Simplify the Formula
As mass \( m \) is present on both sides of the equation, it cancels out, leading to:\[ gh = \frac{1}{2} v^2 \]This simplifies the problem to finding the final velocity \( v \) using the known variables \( g \) and \( h \).
04
Solve for Final Velocity
Rearrange the equation to solve for \( v \):\[ v^2 = 2gh \]\[ v = \sqrt{2gh} \]Substitute in the known values:\[ v = \sqrt{2 \times 9.81 \times 2.31} \]Calculate the value to find \( v \).
05
Calculate the Final Value
Perform the calculation:\[ v = \sqrt{2 \times 9.81 \times 2.31} \approx \sqrt{45.31} \approx 6.73 \text{ m/s} \]
06
Interpret the Result
The calculation shows that the speed of the swimmer at the bottom of the slide is approximately 6.73 meters per second. This result assumes no energy losses to friction or air resistance.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Gravitational Potential Energy
Gravitational potential energy is a form of energy that an object possesses due to its position in a gravitational field. It's the energy stored in an object as it is elevated against gravity. When you lift an object, such as a book or, in the case of our example, when a swimmer ascends a slide, you increase its gravitational potential energy.
The formula to calculate gravitational potential energy is given by:
The formula to calculate gravitational potential energy is given by:
- \( PE = mgh \)
- \( m \) is the mass of the object in kilograms
- \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \) on Earth)
- \( h \) is the height in meters
Kinetic Energy
Kinetic energy refers to the energy that an object possesses due to its motion. The faster an object moves, the more kinetic energy it has. In our water slide scenario, the swimmer's kinetic energy increases as she slides down from the top of the slide.
As the swimmer descends, her gravitational potential energy transforms into kinetic energy. The formula used to represent kinetic energy is:
As the swimmer descends, her gravitational potential energy transforms into kinetic energy. The formula used to represent kinetic energy is:
- \( KE = \frac{1}{2} mv^2 \)
- \( m \) is the mass of the object
- \( v \) is the velocity of the object
Acceleration Due to Gravity
Acceleration due to gravity is a constant that represents the rate at which objects accelerate towards the Earth. It acts on any object that falls or is dropped from a height. On Earth, this value is generally accepted to be approximately \( 9.81 \, \text{m/s}^2 \).
This acceleration is crucial when calculating energy transformations. In the equation \( mgh = \frac{1}{2} mv^2 \), \( g \) is the key factor that affects both the gravitational potential energy and the kinetic energy as the object moves.
In the context of solving our slide problem:
This acceleration is crucial when calculating energy transformations. In the equation \( mgh = \frac{1}{2} mv^2 \), \( g \) is the key factor that affects both the gravitational potential energy and the kinetic energy as the object moves.
In the context of solving our slide problem:
- It ensures that when an object is in free fall or moves without friction, its speed at various points can be predicted accurately using conservation of energy principles.
- Helps in determining how quickly velocity changes as the swimmer descends under the influence of gravity.