Problem 4 - University Physics with Modern Physics

Question

A 72.0 -kg swimmer jumps into the old swimming hole from a diving board 3.25 \(\mathrm{m}\) above the water. Use energy conservation 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 (c) if he manages to jump downward at 2.50 \(\mathrm{m} / \mathrm{s} .\)

Step-by-Step Solution

Answer

(a) 7.99 m/s, (b) 8.37 m/s, (c) 8.80 m/s.

Step 1: Identify Known Values

We have the mass of the swimmer, \( m = 72.0 \) kg, the height of the diving board, \( h = 3.25 \) m, and in cases (b) and (c), the initial velocity \( v_0 = 2.50 \) m/s. The acceleration due to gravity is \( g = 9.81 \) m/s².

Step 2: Write the Energy Conservation Equation

We use the principle of conservation of energy: the initial mechanical energy (potential and kinetic) equals the final kinetic energy when hitting the water. The equation is \( E_i = E_f \), where here \( E_i = mgh + \frac{1}{2}mv_0^2 \) and \( E_f = \frac{1}{2}mv_f^2 \).

Step 3: Solve for Case (a): Drops In

In case (a), the initial kinetic energy is zero (since he just drops), so the equation simplifies to: \( mgh = \frac{1}{2}mv_f^2 \). Solve for \( v_f \) (final speed): \( v_f = \sqrt{2gh} \). Substitute \( g = 9.81 \) m/s² and \( h = 3.25 \) m to find \( v_f = \sqrt{2 \times 9.81 \times 3.25} = \sqrt{63.765} = 7.99 \) m/s.

Step 4: Solve for Case (b): Jumps Upward

Here, the initial kinetic energy is not zero: \( mgh + \frac{1}{2}mv_0^2 = \frac{1}{2}mv_f^2 \). Substitute \( v_0 = 2.50 \) m/s. Thus: \( 72.0 \times 9.81 \times 3.25 + \frac{1}{2} \times 72.0 \times 2.50^2 = \frac{1}{2} \times 72.0 \times v_f^2 \). Solve for \( v_f = \sqrt{2gh + v_0^2} = \sqrt{63.765 + 6.25} = \sqrt{70.015} = 8.37 \) m/s.

Step 5: Solve for Case (c): Jumps Downward

In case (c), modify the initial kinetic term as he's moving downward, thus enhancing initial mechanical energy: \( mgh + \frac{1}{2}mv_0^2 = \frac{1}{2}mv_f^2 \). Thus \( v_f = \sqrt{2gh + v_0^2} = \sqrt{63.765 + 6.25} = \sqrt{70.015} = 8.80 \) m/s.

Key Concepts

Kinetic Energy

Kinetic Energy is the energy possessed by an object due to its motion. It's an essential concept in physics, helping us understand how moving objects interact. The formula for calculating kinetic energy is \[KE = \frac{1}{2}mv^2\]where:

• \(m\) is the mass of the object in kilograms
• \(v\) is the velocity of the object in meters per second

To find the kinetic energy, you need the mass and velocity of the moving objects. The kinetic energy of our swimmer changes based on their initial velocity when diving or jumping.
The energy is initially stored as potential energy, which is then converted to kinetic energy as the swimmer falls. This conversion process is a key aspect of energy conservation.

Potential Energy

Potential Energy is stored energy based on an object's position. In the case of our swimmer, it's the energy stored due to his height above the water. The formula for gravitational potential energy is:\[PE = mgh\]where:

• \(m\) is the object's mass in kilograms
• \(g\) is the acceleration due to gravity, approximately 9.81 m/s²
• \(h\) is the height above the reference point, in meters

When the swimmer is on the diving board, he has potential energy due to his elevation. As he jumps, this potential energy is gradually converted to kinetic energy as he falls. Understanding how potential energy transforms helps in solving physics problems, especially those involving movement and height. It provides insight into how and why objects move the way they do.
In simple scenarios like this, potential energy plays a crucial role in calculations involving energy conservation.

Physics Problem Solving

Physics Problem Solving requires systematically applying fundamental concepts to find solutions. When tackling problems involving Conservation of Energy, you should:

• Identify known and unknown variables: Start by determining what information you have and what you need to find.
• Apply relevant principles: Use the principle of energy conservation, which states that energy in a closed system remains constant.
• Break down the problem: Analyze each step logically, such as calculating potential and kinetic energy and how they transform.
• Formulate equations: Use standard equations related to potential and kinetic energy.
• Calculate accurately: Pay attention to units and carefully perform calculations to find results like final velocity or energy values.

In our exercise, the conservation of energy was used to understand how different initial conditions affected the swimmer's speed upon hitting the water. Each case required careful consideration of initial conditions, energy transformations, and precise calculations. This approach is key in traversing even the most daunting physics problems successfully.