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Chapter 5: Problem 3

An engine pumps water continuously through a hose. If the speed with which the water passes through the hose nozzle is \(v\) and if \(k\) is the mass per unit length of the water jet as it leaves the nozzle, what is the kinetic energy being imparted to the water? a) \(\frac{1}{2} k v^{3}\) c) \(\frac{1}{2} k v\) e) \(\frac{1}{2} v^{3} / k\) b) \(\frac{1}{2} k v^{2}\) d) \(\frac{1}{2} v^{2} / k\)

Short Answer

Expert verified
Answer: (b) \(\frac{1}{2} k v^{2}\)

Step by step solution

01

Remember the Kinetic Energy formula

The formula for kinetic energy (KE) is KE = \(\frac{1}{2}mv^2\), where m is the mass of the object and v is its speed.
02

Calculate the mass of the water in terms of k

We are given k as the mass per unit length (mass/length) of the water jet. To find the total mass, multiply k by the length (L) of the water jet, which gives us m = kL.
03

Substitute mass into the Kinetic Energy formula

Now, we substitute m = kL into the KE formula: KE = \(\frac{1}{2}(kL)v^2\)
04

Simplify the expression

Simplify the expression for KE: KE = \(\frac{1}{2}kLv^2\)
05

Compare the expression with the options

Compare our derived expression, \(\frac{1}{2}kLv^2\), with the given options: a) \(\frac{1}{2} k v^{3}\) c) \(\frac{1}{2} k v\) e) \(\frac{1}{2} v^{3} / k\) b) \(\frac{1}{2} k v^{2}\) d) \(\frac{1}{2} v^{2} / k\)
06

Identify the correct option

Option (b) \(\frac{1}{2} k v^{2}\) matches our derived expression for the kinetic energy being imparted to the water, so the correct answer is: (b) \(\frac{1}{2} k v^{2}\)

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Most popular questions from this chapter

You push your couch a distance of \(4.00 \mathrm{~m}\) across the living room floor with a horizontal force of \(200.0 \mathrm{~N}\). The force of friction is \(150.0 \mathrm{~N}\). What is the work done by you, by the friction force, by gravity, and by the net force?

If the net work done on a particle is zero, what can be said about the particle's speed?

A certain tractor is capable of pulling with a steady force of \(14 \mathrm{kN}\) while moving at a speed of \(3.0 \mathrm{~m} / \mathrm{s}\). How much power in kilowatts and in horsepower is the tractor delivering under these conditions?

The mass of a physics textbook is \(3.4 \mathrm{~kg}\). You pick the book up off a table and lift it \(0.47 \mathrm{~m}\) at a constant speed of \(0.27 \mathrm{~m} / \mathrm{s}\) a) What is the work done by gravity on the book? b) What is the power you supplied to accomplish this task?

An arrow of mass \(m=88 \mathrm{~g}(0.088 \mathrm{~kg})\) is fired from a bow. The bowstring exerts an average force of \(F=110 \mathrm{~N}\) on the arrow over a distance \(d=78 \mathrm{~cm}(0.78 \mathrm{~m})\) Calculate the speed of the arrow as it leaves the bow.
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