Question

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

The swimmer's speed at the bottom is 6.73 m/s.

Step-by-step solution

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/s².

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.

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 \( h \).

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 \( v \).

Calculate the Final Value

Perform the calculation:\[ v = \sqrt{2 \times 9.81 \times 2.31} \approx \sqrt{45.31} \approx 6.73 \text{ m/s} \]

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.

Key concepts

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:

• \( PE = mgh \)

where:

• \( m \) 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

In the absence of external forces like friction, the potential energy will convert entirely into other forms of energy, like kinetic energy, as the object descends. This principle is fundamental in problems involving the conservation of energy.

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:

• \( KE = \frac{1}{2} mv^2 \)

where:

• \( m \) is the mass of the object
• \( v \) is the velocity of the object

In situations where friction is negligible, like in our problem, the potential energy at the top converts fully into kinetic energy at the bottom. This results in the swimmer gaining maximum possible kinetic energy, which corresponds to her speed when reaching the pool.

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:

• 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.

Understanding this concept allows us to make precise calculations about how energy shifts from potential to kinetic, and finally, allows us to find out the speed with which the swimmer enters the pool.