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Chapter 6: Problem 58

Calculate A kayaker paddles with a power output of \(50.0 \mathrm{~W}\) to maintain a steady speed of \(1.50 \mathrm{~m} / \mathrm{s}\). Find the force exerted by the kayaker.

Short Answer

Expert verified
The force exerted by the kayaker is 33.3 N.

Step by step solution

01

Understand the Power Formula

The power output of the kayaker is given by the formula: \[ P = F \times v \]where \( P \) is the power, \( F \) is the force, and \( v \) is the velocity.
02

Rearrange the Formula to Solve for Force

We need to rearrange the formula to find the force \( F \). This can be done by dividing both sides by the velocity \( v \):\[ F = \frac{P}{v} \]
03

Substitute the Given Values

Substitute the given values into the rearranged formula. We have the power \( P = 50.0 \mathrm{~W} \) and the velocity \( v = 1.50 \mathrm{~m/s} \):\[ F = \frac{50.0 \mathrm{~W}}{1.50 \mathrm{~m/s}} \]
04

Calculate the Force

Perform the division to calculate the force:\[ F = \frac{50.0}{1.50} = 33.3 \mathrm{~N} \]

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Key Concepts

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

Power Formula
In physics, power is a measure of how much work is done by a force within a certain time frame. The Power Formula helps us to connect power, force, and velocity through a simple equation:
  • Power ( P ) is equal to force ( F ) times velocity ( v ), represented as: \[ P = F \times v \]
This formula is crucial when dealing with problems related to moving objects. The power value reflects the rate at which energy is used or transferred. If we know the power and velocity, we can rearrange the formula to solve for force. This makes it easier to determine how much effort or force is applied to achieve a certain speed. Breaking down equations is a fundamental skill in problem-solving, allowing you to isolate the desired variable.
Force Calculation
Force is a critical concept in physics, representing the interaction that changes the motion of an object. In our scenario, the kayaker's purpose is to understand what force is needed to maintain a steady speed.To calculate this, we use the rearranged Power Formula:
  • \[ F = \frac{P}{v} \]
By dividing the power by velocity, we isolate the force. This equation shows that force is the power distributed over a specific pace or speed. When substituting real numbers into this equation, we found the kayaker exerts a force of 33.3 Newtons to paddle at a constant velocity. This step is about understanding the relationship between force, power, and speed.
Velocity
Velocity refers to the speed of an object in a specified direction. It is a vector quantity, meaning it conveys both how fast something is moving and its direction. For the kayaker, velocity is the steady speed of 1.50 meters per second. Velocity is an integral part of the Power Formula because it dictates how much power will be transformed into consistent movement (force). Understanding the concept of velocity in physics helps students grasp the dynamics involved in motion.
Essentially, if you know the force applied and the resulting velocity, you can determine the efficiency of the action performed, like the kayaker's paddling in our example.
Kayak Physics
Kayak Physics revolves around understanding how physical principles like power, velocity, and force interact when paddling. A kayaker uses energy, expressed as power, to propel the kayak forward. The case of the kayaking problem involves:
  • Applying force through paddling;
  • Maintaining a constant velocity, requiring enough force to counteract water resistance;
  • Measuring how much power is necessary to achieve this motion.
In practical terms, the force exerted by the kayaker is directed against the resistance of water. This requires ample energy, translated into power, to maintain motion. This problem demonstrates not only how physical formulas are applied but how input energy affects movement when battling elements like drag and water currents.

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Most popular questions from this chapter

After hitting a long fly ball that goes over the right fielder's head and lands in the outfield, a batter decides to keep going past second base and try for third base. The \(62-\mathrm{kg}\) player begins sliding \(3.4 \mathrm{~m}\) from the base with a speed of \(4.5 \mathrm{~m} / \mathrm{s}\). (a) If the player comes to rest at third base, how much work was done on the player by friction with the ground? (b) What was the coefficient of kinetic friction between the player and the ground?

A weightlifter does \(9.8 \mathrm{~J}\) of work while lifting a weight straight upward through a distance of \(0.12 \mathrm{~m}\). What was the force exerted by the weightlifter?

Research and write a report on the power that humans produce in everyday life. Include the power produced by the brain when thinking and the heart when resting. Also select several strenuous activities such as trackand-field events, bicycle racing, and swimming. In each case, explain how the power is determined. Make a table to compare the various power outputs.

A crow drops a \(0.11-\mathrm{kg}\) clam onto a rocky beach from a height of \(9.8 \mathrm{~m}\). What is the kinetic energy of the clam when it is \(5.0 \mathrm{~m}\) above the ground? What is its speed at that point?

A spring has a spring constant of \(310 \mathrm{~N} / \mathrm{m}\). Plot the potential energy for this spring when it is stretched by \(1.0 \mathrm{~cm}, 2.0 \mathrm{~cm}, 3.0 \mathrm{~cm}\), and \(4.0 \mathrm{~cm}\). Draw a curve that goes through your plotted points.
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