Question

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 approximately 6.72 m/s.

Step-by-step solution

Understand the Principle

This problem can be solved using the principle of conservation of mechanical energy, as there is no friction. The swimmer's potential energy at the top converts into kinetic energy at the bottom.

Identify Initial and Final Energies

Initially, all the energy in the system is potential energy (PE) because the swimmer starts at rest. The potential energy at the top is given by the formula \( PE = mgh \), where \( m \) is mass, \( g \) is the acceleration due to gravity \( (9.81 \ \text{m/s}^2) \), and \( h \) is the height (2.31 meters). The final energy is kinetic, given by \( KE = \frac{1}{2}mv^2 \).

Set Up Conservation Equation

According to the conservation of energy, the potential energy at the top equals the kinetic energy at the bottom. Therefore, the equation is \( mgh = \frac{1}{2}mv^2 \).

Cancel Mass and Solve for Velocity

Since the mass \( m \) appears on both sides of the equation, it can be canceled out. simplifying to \( gh = \frac{1}{2}v^2 \). Solving for \( v \), we have \( v = \sqrt{2gh} \).

Substitute Values and Calculate

Substitute \( g = 9.81 \ \text{m/s}^2 \) and \( h = 2.31 \ \text{m} \) into the equation: \( v = \sqrt{2 \times 9.81 \times 2.31} \). Calculating this gives \( v \approx 6.72 \ \text{m/s} \).

Key concepts

Potential Energy

Potential energy is the energy that an object possesses due to its position in a gravitational field. Imagine a swimmer at the top of a water slide; here, they have a certain potential energy. This energy depends on how high up the slide they are positioned.
The formula to calculate potential energy is simple: \( PE = mgh \). In this formula:

• \( m \) stands for mass.
• \( g \) is the gravitational acceleration (usually \(9.81 \, \text{m/s}^2\) on Earth).
• \( h \) is the height from which the swimmer starts.

Potential energy is about stored energy that could potentially be transformed into other forms, like kinetic energy, once movement begins. So, if our swimmer starts sliding down, this stored potential energy will convert into movement, translating into kinetic energy.

Kinetic Energy

Kinetic energy is the energy of motion. When our swimmer begins sliding down the water slide, their potential energy gradually converts into kinetic energy.
This transformation can be calculated using the formula: \( KE = \frac{1}{2}mv^2 \). In this equation:

• \( m \) is the mass of the swimmer.
• \( v \) represents their velocity at any given point in motion.

As the swimmer reaches the bottom of the slide, the speed has increased due to gravitational pull. The initial potential energy is now fully changed into kinetic energy. In a frictionless system, like our ideal slide, this interchange between energies occurs seamlessly. The equation \( mgh = \frac{1}{2}mv^2 \) shows how potential energy at the top is equal to kinetic energy at the bottom when friction isn’t a factor.

Frictionless Motion

In the context of physics problems, frictionless motion refers to an ideal scenario where there is no friction between the moving object and the surface. On a frictionless slide, the conversion of energy follows a perfect model where no energy is lost to heat or other forces.
This is key because it means that all potential energy is transferred to kinetic energy. In the real world, some energy would be lost due to friction, slightly reducing the final speed of the swimmer at the bottom of the slide. But in our scenario, friction doesn't play a role.
This ideal condition helps simplify calculations. Students can focus purely on the transformation of energy without factoring in energy losses. Therefore, it highlights the fundamental principle of conservation of mechanical energy, where the total mechanical energy remains constant if only conservative forces like gravity are at work.