Momentum and Newton's Laws of Motion

# Jet Force

A jet of water from a high-lying lake sprays at high speed onto the blades of a Pelton turbine wheel (see Figure 1, left). Due to the high speed, the force of the jet on the blades is also high. The wheel rotates at speed and drives e.g., an electric generator.

The blades of a Pelton turbine are designed so that the water - after giving off energy to the turbine - bounces back in the direction from which it acted on the turbine (see Figure 1, right). We will learn why this is necessary in the next chapter.

Figure 1: Force of water jet on Pelton turbine

When we talk about jet force, we mean the force of a jet of any liquid: water or gas (e.g. air).

Example

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But what is the physical background of the force with which the jet acts on a body and the body back on the jet?

Moving water has its own mass and speed. So it has momentum. When it hits an obstacle, it can bounce off the obstacle as in the case of a Pelton turbine, or it can lose its entire momentum and drain to the ground. In both cases, the obstacle acts on the jet (or the jet on the obstacle - see Interaction of forces ) and changes its momentum.

## Volume flow rate

A jet can be thought of as a volumetric flow of matter. Take, for example, a cylindrical section of such a jet. Let's imagine a base of surface area moving along with the jet (see Figure 2). In time , the cross-sectional area makes a distance , in this case, the cross-sectional area describes the volume .

Figure 2: Volume flow rate

the volume flow rate is defined as the change in volume (or elapsed volume) in the observed time :

 We note that: We note also that:

The volume flow rate is the change in volume during the observed time:

It can be calculated if we know the cross-sectional area and the velocity of the jet:

Example

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## Mass flow rate

If we were interested in the previous chapter:

• the volume that flows in a given time,

we must also be interested in:

• the mass that flows in a given time.

Let some mass move along a cylindrical tube. Mass flow rate is defined as the ratio of the:

• mass passing through the observed cross-section of the tube at the observed time and

• observed time .

We write the equation:

 We note that the mass can be expressed in terms of the density and volume as: or since we are talking about changes: We note from above that: We note that:

The mass flow rate tells how much mass of a substance flows past in a certain time:

It can be expressed in terms of the volume flow rate and the density of the substance:

or in terms of the density of the substance, cross-sectional area and velocity of the jet:

Example

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Consider a jet flowing through a pipe of cross-sectional area with velocity . The tube gradually narrows to a cross-sectional area . In order for the same volume of substance to pass in the same time in both parts of the pipe, we expect that the substance will have to move faster in the narrower part than in the wider one.

Steady flow is a flow that always has the same volume flow, regardless of where it is measured

If the flow is steady, the volume flow rate in both parts of the pipe must be the same:

 We note that: Therefore:

In the case of a steady flow, the product of the cross-sectional area and velocity of the jet is constant:

Example

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## Jet force

The jet hits an obstacle and bounces off it. The obstacle acts on the jet with a force impulse by changing its momentum. Let's observe the impact of the force on a small mass that is in contact with the obstacle during for time :

 We divide both sides of the equation by : We note that: We note that: We note that the mass flow rate of the jet before colliding with an obstacle is given as:

Due to the mutual interaction of forces, the jet acts on the obstacle with an opposite equal force:

The force of the jet is the product of the mass flow rate and the difference in its velocities after the collision and before the collision:

where:

The force of the jet is the opposite of the force with which the obstacle acts on the jet:

Example

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## Reaction force

Let a jet of water emerge from a nozzle. According to the interaction of forces, a force opposite to the jet force acts on the nozzle itself by pushing it in the opposite direction as the jet velocity. The force is called the reaction force.

Let us assume that the jet of water comes out of the nozzle with a velocity . The reaction force is given as:

 We note that:

The reaction force is directed in the opposite direction to the jet flow and is the product of the mass flow rate and the velocity of the jet:

If a jet of a substance of density , cross-sectional area and speed comes out of a body, the reaction force is given as:

Example

The example is available to registered users free of charge.

Example

The example is available to registered users free of charge.

material editor: Gabriel Amakhabi