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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

An engine expends 40.0 hp in moving a car along a level track at a speed of \(15.0 \mathrm{~m} / \mathrm{s}\). How large is the total force acting on the car in the opposite direction of the motion of the car?

A car, of mass \(m,\) traveling at a speed \(v_{1}\) can brake to a stop within a distance \(d\). If the car speeds up by a factor of \(2, v_{2}=2 v_{1},\) by what factor is its stopping distance increased, assuming that the braking force \(F\) is approximately independent of the car's speed?

A spring with a spring constant of \(238.5 \mathrm{~N} / \mathrm{m}\) is compressed by \(0.231 \mathrm{~m}\). Then a steel ball bearing of mass \(0.0413 \mathrm{~kg}\) is put against the end of the spring, and the spring is released. What is the speed of the ball bearing right after it loses contact with the spring? (The ball bearing will come off the spring exactly as the spring returns to its equilibrium position. Assume that the mass of the spring can be neglected.)

While a boat is being towed at a speed of \(12 \mathrm{~m} / \mathrm{s}\), the tension in the towline is \(6.0 \mathrm{kN}\). What is the power supplied to the boat through the towline?

A spring with spring constant \(k\) is initially compressed a distance \(x_{0}\) from its equilibrium length. After returning to its equilibrium position, the spring is then stretched a distance \(x_{0}\) from that position. What is the ratio of the work that needs to be done on the spring in the stretching to the work done in the compressing?
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