Problem 39 - Algebra and Trigonometry

Question

Drag Force on a Boat The drag force \(F\) on a boat is jointly proportional to the wetted surface area \(A\) on the hull and the square of the speed \(s\) of the boat. A boat experiences a drag force of 220 lb when traveling at 5 mi/h with a wetted surface area of 40 \(\mathrm{ft}^{2}\) . How fast must a boat be traveling if it has 28 \(\mathrm{ft}^{2}\) of wetted surface area and is experiencing a drag force of 175 \(\mathrm{lb} ?\)

Step-by-Step Solution

Answer

The boat must travel at approximately 5.33 mi/h.

Step 1: Understand the Relationship

The drag force \(F\) is described as being jointly proportional to the wetted surface area \(A\) and the square of the speed \(s\) of the boat. This means we can express the relationship as: \(F = k \cdot A \cdot s^2\), where \(k\) is a proportionality constant.

Step 2: Determine the Proportionality Constant

Given that for a drag force of 220 lb, \(A = 40 \ \mathrm{ft}^2\) and \(s = 5 \ \mathrm{mi/h}\), substitute these values into the equation: \(220 = k \cdot 40 \cdot 5^2\). Simplifying gives \(220 = k \cdot 40 \cdot 25\) which further reduces to \(220 = 1000k\), and hence \(k = \frac{220}{1000} = 0.22\).

Step 3: Set Up the Equation for the New Condition

For the new scenario, where \(F = 175 \ \mathrm{lb}\) and \(A = 28 \ \mathrm{ft}^2\), we need to find \(s\). Use the known \(k\) to set up the equation: \(175 = 0.22 \cdot 28 \cdot s^2\).

Step 4: Solve for the Speed

From the equation \(175 = 0.22 \cdot 28 \cdot s^2\), compute \(0.22 \cdot 28 = 6.16\). Then replace in the equation: \(175 = 6.16 \cdot s^2\). Now solve for \(s^2\): \(s^2 = \frac{175}{6.16}\). This simplifies to \(s^2 \approx 28.4091\). Take the square root to find \(s\): \(s \approx 5.33 \ \mathrm{mi/h}\).

Key Concepts

Proportionality Constant

When we talk about joint variation, like in the case of drag force on a boat, the concept of a proportionality constant is crucial. Here, the drag force \( F \) is jointly proportional to the wetted surface area \( A \) and the square of the speed \( s \). We express this relationship with the equation: \[ F = k \cdot A \cdot s^2 \] Here, \( k \) is what we refer to as the proportionality constant.

• This constant \( k \) helps us connect the various factors (\( A \) and \( s^2 \)) to the drag force \( F \).
• The value of \( k \) remains the same even if the specific values for \( A \) or \( s \) change, as long as the conditions are similar.

In real-life applications, once\( k \) is determined using known values (like in the worked-out example), it can predict unknown quantities like the speed of the boat when other variables change.

Drag Force

Drag force is a resistance force caused by the motion of the boat through water. It is an important factor to consider in marine engineering and physics. The drag force increases as either the wetted surface area or the speed of the boat increases. Described by the equation \( F = k \cdot A \cdot s^2 \), this relationship illustrates that:

• The larger the surface area in contact with the water, the greater the resistance or drag force.
• Similarly, the faster the boat moves, the higher the resistance, due to the \( s^2 \) term.

Understanding drag force helps in designing efficient boats that can move swiftly through water while managing fuel consumption and energy efficiency.

Surface Area and Speed Relationship

The wetted surface area and the speed of a boat have a direct relationship in creating drag force. These factors multiply to determine how much resistance the boat encounters. We find this through the equation \( F = k \cdot A \cdot s^2 \). Here's how they interact:

• The wetted surface area \( A \) is the part of the boat's hull submerged in water. A larger surface area means more friction with the water, increasing drag force.
• The speed \( s \) of the boat enters the equation as \( s^2 \), meaning any increase in speed significantly raises the drag force because it is squared.

By understanding this relationship, boat designers can manipulate either the surface area or manage the boat's speed to optimize performance and efficiency.