Question

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

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

Step-by-step solution

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.

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.

Substitute mass into the Kinetic Energy formula

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

Simplify the expression

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

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) \(\frac{1}{2} v^{2} / k\)

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) \(\frac{1}{2} k v^{2}\)

Key concepts

Mass Per Unit Length

Understanding the concept of mass per unit length is crucial when dealing with problems involving continuous materials like water jets, cables, or rods.Unlike in rigid bodies where the mass is fixed and specific, in a continuous medium, mass is distributed along a length. This means that for any given length of the material, you have a corresponding mass associated with that portion. For example, with a water jet, if you consider a section of the jet that is one meter long, and it weighs 2 kilograms, then the mass per unit length would be 2 kilograms per meter. This is denoted by the variable \(k\), often used in such calculations.

• Mass per unit length \(k\) = total mass / length
• For a water jet, this helps in calculating total mass when length is known.

When working with kinetic energy for continuous streams, like water jets, knowing the mass per unit length allows you to express mass in terms of measurable quantities (like length), making it easier to apply formulas and principles.

Kinetic Energy Formula

Kinetic energy is a critical concept in physics, representing the energy possessed by an object due to its motion.The formula for kinetic energy is given by:\[ KE = \frac{1}{2}mv^2 \]where \(m\) is the mass of the object and \(v\) is its velocity. This formula is essential when calculating the energy of moving objects, whether they be cars, tennis balls, or even a stream of water.

• \(m\) stands for mass, indicating how much matter is present.
• \(v\) represents velocity, showing how fast the object is moving.
• The factor \(\frac{1}{2}\) arises from the mathematical derivation of kinetic energy, often involving calculus and work-energy principles.

For the water jet scenario, we replace the object’s mass \(m\) with \(kL\), where \(k\) is the mass per unit length and \(L\) is the length of the jet. So the formula suited to a water jet becomes \[ KE = \frac{1}{2}kLv^2 \]. This is how we calculate the energy of water being expelled at a certain speed from a nozzle.

Speed of Water Jet

The speed of a water jet is a crucial factor in determining the energy dynamics of the water being expelled from a nozzle. When we say that a water jet has speed \(v\), we refer to how fast the water is moving as it exits the nozzle. The speed impacts the kinetic energy since it appears in the formula squared, amplifying the effect of speed on energy. A doubling of speed, for instance, results in four times the kinetic energy.

• Measured in units like meters per second (m/s).
• Serves as a key determinant for the jet's kinetic energy.

Understanding the speed of the water jet provides insight into its potential to do work, such as moving an object or generating power. This makes it a fundamental variable in fluid dynamics and energy calculations.