Problem 32 - College Physics

Question

\(\bullet\) A 72.0 -kg swimmer jumps into the old swimming hole from a diving board 3.25 m above the water. Use energy conserva- tion to find his speed just he hits the water (a) if he just holds his nose and drops in, (b) if he bravely jumps straight up (but just beyond the board!) at \(2.50 \mathrm{m} / \mathrm{s},\) and \((\mathrm{c})\) if he manages to jump downward at 2.50 \(\mathrm{m} / \mathrm{s}\) .

Step-by-Step Solution

Answer

(a) 7.98 m/s, (b) 8.33 m/s, (c) 8.59 m/s.

Step 1: Understanding Energy Conservation

We start by identifying that the swimmer converts potential energy (due to height) into kinetic energy (due to speed) as they fall. The total mechanical energy is conserved (if we neglect air resistance). The key equation here is \( mgh = \frac{1}{2}mv^2 \).

Step 2: Calculate Potential Energy at the Start

Potential energy at the start is given by \( PE = mgh \), where \( m \) is the mass (72.0 kg), \( g \) is the acceleration due to gravity (9.8 m/s²), and \( h \) is the height (3.25 m). Substitute to find \( PE = 72 \times 9.8 \times 3.25 \).

Step 3: Calculate Speed for Part (a): Just Holds His Nose and Drops

For part (a), the swimmer just drops, so initial kinetic energy is zero. Set potential energy equal to kinetic energy: \( mgh = \frac{1}{2}mv^2 \). Solve for \( v \): \( v = \sqrt{2gh} \). Substitute the values: \( v = \sqrt{2 \times 9.8 \times 3.25} \) to find the speed.

Step 4: Calculate Speed for Part (b): Jumps Straight Up

For part (b), the swimmer jumps up with an initial speed. Use \( K.E._{initial} = \frac{1}{2}mv_0^2 \) (\( v_0 = 2.5 \mathrm{m/s} \)): \( v^2 = v_0^2 + 2gh \). Substitute to calculate: \( v^2 = (2.50)^2 + 2 \times 9.8 \times 3.25 \) and solve for \( v \).

Step 5: Calculate Speed for Part (c): Jumps Downward

For part (c), the swimmer jumps downward initially, which adds to the potential energy conversion. Use \( K.E._{initial} = \frac{1}{2}mv_0^2 \): \( v^2 = v_0^2 + 2gh \). Here, \( v_0 = 2.5 \mathrm{m/s} \) downward. Substitute to calculate \( v^2 = (2.50)^2 + 2 \times 9.8 \times 3.25 \) and solve for \( v \).

Key Concepts

Potential Energy

Potential energy is stored energy that depends on an object's position or condition. In the context of our swimming exercise, it is the energy possessed by the swimmer due to their height above the water. Potential energy (PE) can be calculated using the formula:

• \( PE = mgh \)

where:

• \( m \) is the mass of the object, which is 72.0 kg for the swimmer,
• \( g \) is the acceleration due to gravity, approximately 9.8 m/s² on Earth,
• \( h \) is the height above the water, given as 3.25 m.

By substituting these values into the formula, you can determine the amount of stored energy before the swimmer dives. This energy is crucial since it gets transformed into kinetic energy when the swimmer jumps, showing the conservation of energy in action.

Kinetic Energy

Kinetic energy refers to the energy an object possesses due to its motion. When the swimmer in our exercise moves from the diving board to the water, potential energy transforms into kinetic energy. The kinetic energy (KE) of the swimmer at any point during the dive can be expressed as:

• \( KE = \frac{1}{2}mv^2 \)

where:

• \( m \) is the mass of the swimmer (72.0 kg),
• \( v \) is the velocity or speed of the swimmer at that moment.

The process of conversion from potential to kinetic energy explains why the swimmer gains speed as they descend. The higher the initial height, the more potential energy, and thus more kinetic energy can be obtained, leading to a quicker descent as more energy is converted.

Mechanical Energy

Mechanical energy is the sum of an object's potential and kinetic energy. It represents the total energy available for the swimmer's motion and is governed by the principle of energy conservation. In our exercise, mechanical energy remains constant if we ignore air resistance. This means the sum of potential and kinetic energy at any point in the swimmer's dive equals the total mechanical energy initially present.Let's simplify things with an equation:

• \( ME = PE + KE \)

At the top of the diving board, the swimmer's mechanical energy is all potential (since speed and thus kinetic energy are zero). As they fall, potential energy decreases while kinetic energy increases, keeping the mechanical energy the same.This principle helps calculate how fast the swimmer will move at different points, allowing us to solve each part of the exercise accurately by balancing potential and kinetic energy in the calculations.