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Chapter 8: Problem 99

A swimmer moves through the water at an average speed of \(0.22 \mathrm{~m} / \mathrm{s}\). The average drag force is \(110 \mathrm{~N}\). What average power is required of the swimmer?

Short Answer

Expert verified
The average power required is 24.2 watts.

Step by step solution

01

Understand the Formula for Power

The formula for power (P) when force (F) and velocity (v) are involved is given by \( P = F \cdot v \). Power is defined as the rate at which work is done, and in this case, it's the product of the force exerted by the swimmer against the drag and their velocity in the water.
02

Identify Known Values

From the problem, we know the average speed of the swimmer is \( v = 0.22 \text{ m/s} \) and the average drag force is \( F = 110 \text{ N} \). These values will be used in the formula for power.
03

Calculate the Average Power

Substitute the known values into the power formula: \( P = 110 \text{ N} \times 0.22 \text{ m/s} \). This gives \( P = 24.2 \text{ watts} \). This is the average power required to overcome the drag force while moving at the given speed.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Drag Force
Drag force is the resistance an object encounters as it moves through a fluid, such as air or water. In our example, the drag force on the swimmer is caused by water resistance. Understanding the factors impacting drag force is essential for calculating how much effort is required to keep moving. Drag force depends on several key elements:
  • The speed of the object: Faster movement typically increases the drag.
  • The shape and surface area: A streamlined shape can reduce drag.
  • The density of the fluid: Water generally offers more resistance than air.
In the case of our swimmer, the drag force exerted by the water is 110 N. This means they have to exert an equivalent amount of force to maintain their speed.
Velocity
Velocity is a vector quantity that refers to the speed of an object in a specified direction. Understanding velocity is crucial for determining the power needed for movement in fluid dynamics like swimming. In this example, the swimmer maintains a velocity of 0.22 m/s. Velocity can be affected by:
  • Force applied: More force can increase velocity, assuming other conditions are constant.
  • Resistance encountered: Higher resistance (like drag force) may decrease velocity.
Knowing the swimmer's velocity allows us to calculate how quickly they can cover a given distance, as well as determine the power requirements needed.
Power Formula
The power formula is a mathematical equation used to determine the rate at which work is done. It combines force and velocity to find out how much power is required for movement. The formula is given by:\[ P = F \cdot v \]Where:
  • P is power in watts.
  • F is force in newtons.
  • v is velocity in meters per second.
In this swimming example, the problem asks us for the power required to keep moving through water with a given drag force and velocity. By plugging in F = 110 N and v = 0.22 m/s, we find that the average power needed is 24.2 watts. This formula enables swimmers and engineers to calculate exactly how much effort is needed to maintain a certain speed against resistance.
Work Done
Work done is a measure of the energy transferred when a force is applied to an object over a distance. In the context of power and drag force, it helps us understand how much energy a swimmer uses when swimming.Work done can be calculated with the formula:\[ W = F \cdot d \]Where:
  • W is work done in joules.
  • F is force in newtons.
  • d is the distance in meters.
Although work done is not specified in this problem, it is related to power as power is the rate of doing work. Knowing the work done over a period can aid in understanding overall energy requirements. In the swimmer's scenario, consistent work is done to overcome the drag force, enabling the swimmer to keep moving forward.

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Most popular questions from this chapter

Approximately \(5.5 \times 10^{6} \mathrm{~kg}\) of water falls \(50 \mathrm{~m}\) over Niagara Falls each second. (a) What is the decrease in the gravitational potential energy of the water-Earth system each second? (b) If all this energy could be converted to electrical energy (it cannot be), at what rate would electrical energy be supplied? (The mass of \(1 \mathrm{~m}^{3}\) of water is \(1000 \mathrm{~kg} .\) ) (c) If the electrical energy were sold at 1 cent \(/ \mathrm{kW} \cdot \mathrm{h},\) what would be the yearly income?

A \(9.40 \mathrm{~kg}\) projectile is fired vertically upward. Air drag decreases the mechanical energy of the projectile Earth system by \(68.0 \mathrm{~kJ}\) during the projectile's ascent. How much higher would the projectile have gone were air drag negligible?

The only force acting on a particle is conservative force \(\vec{F}\). If the particle is at point \(A,\) the potential energy of the system associated with \(\vec{F}\) and the particle is \(40 \mathrm{~J}\). If the particle moves from point \(A\) to point \(B,\) the work done on the particle by \(\vec{F}\) is \(+25 \mathrm{~J}\). What is the potential energy of the system with the particle at \(B ?\)

A river descends \(15 \mathrm{~m}\) through rapids. The speed of the water is \(3.2 \mathrm{~m} / \mathrm{s}\) upon entering the rapids and \(13 \mathrm{~m} / \mathrm{s}\) upon leaving. What percentage of the gravitational potential energy of the water-Earth system is transferred to kinetic energy during the descent? (Hint: Consider the descent of, say, \(10 \mathrm{~kg}\) of water.)

In Fig. 8 -42, a block of mass \(m=3.20 \mathrm{~kg}\) slides from rest a distance \(d\) down a frictionless incline at angle \(\theta=30.0^{\circ}\) where it runs into a spring of spring constant \(431 \mathrm{~N} / \mathrm{m} .\) When the block momentarily stops, it has compressed the spring by \(21.0 \mathrm{~cm} .\) What are (a) distance \(d\) and (b) the distance between the point of the first block-spring contact and the point where the block's speed is greatest?
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