Question

A swimmer moves through the water at a speed of \(0.22 \mathrm{~m} / \mathrm{s}\). The drag force opposing this motion is \(110 \mathrm{~N}\). How much power is developed by the swimmer?

Short answer

The power developed by the swimmer is \(24.2 watts\).

Step-by-step solution

Identify the given values

From the problem we can identify the following values: velocity of the swimmer \(v = 0.22 m/s\) and the drag force \(F = 110 N\).

Use the formula for power

The formula to calculate power in physics is \(Power = Force \times Velocity\).

Substitute the values into the formula

Substitute the given values into the formula: \[Power = 110 N \times 0.22 m/s\].

Calculate the power

By multiplying the force and velocity, you obtain a power of \(24.2 watts\).

Key concepts

Work-Energy Principle

The work-energy principle is a fundamental concept in physics that relates the work done on an object to the change in its energy. In the context of the problem at hand, the work performed by the swimmer against the drag force results in kinetic energy that propels them through the water.

Work is defined as the product of force and displacement in the direction of the force: \( Work = Force \times Displacement \). When a force is applied over a distance, energy is transferred and work is done. Understanding this principle is crucial in calculating the swimmer's power output, as the power is directly related to the work done per unit of time.

In simpler terms, if our swimmer applies a force to overcome the resistance of the water, the energy used to do this work is manifested in the swimmer's forward motion. If the force and the motion are constant, we can then link this to the power generated by the swimmer, who uses energy at a certain rate to maintain their speed.

Drag Force

Drag force is the resistance an object encounters as it moves through a fluid, such as water or air. This force opposes the motion and is influenced by factors like the object's shape, speed, and the fluid's properties.

The swimmer in our exercise experiences drag force from the water, which tries to slow them down. Calculating drag force can be complex, involving fluid dynamics and surface area, but in this context, the drag force is already given as \(110 N\). For the swimmer to maintain a constant speed, they must exert enough force to counteract this drag. The exertion against drag represents energy consumption, linking back to the work done per time, which we express as power.

Understanding drag force is essential as it helps us understand how much force the swimmer needs to exert to maintain their movement through the water. It's this continuous 'fight' against drag that contributes significantly to the power the swimmer develops.

Power Formula

The power formula in physics is a crucial tool for quantifying the rate at which work is done or energy is transferred. It is expressed as \(Power = Force \times Velocity\), where force is the constant push or pull acting along the direction of movement, and velocity is the speed in that same direction.

Returning to our example, the power developed by the swimmer can be calculated by multiplying the drag force they work against (\(110 N\)) by their constant speed (\(0.22 m/s\)), resulting in an output of \(24.2 watts\). This calculation reveals that the swimmer is using energy at a rate of \(24.2 watts\) to overcome the resistance of the water and maintain their speed.

It's worth noting that the power output conveys how much work the swimmer does in a given time. So, a higher power output means more work is done quickly, which translates to greater energy expenditure. This concept is not only essential for solving physics problems but also has practical applications in everyday life and various industries where understanding energy transfer rates is vital.