Velocity and Acceleration

 

Velocity Components

 

Horizontal velocity is a measure ofthe horizontal displacement of a fluid parcel.  If “s” is a symbol for position, and “t” is a symbol for time, then the horizontal velocity is

 

 

These are symbols you are probably familiar with from high school physics.  But they are also symbols used in one of the systems we use to locate ourselves relative to the earth or to a map or to characterize flow. These systems are known as “coordinate” systems.   The symbols used above are used in the “natural coordinate” system.

 

A coordinate system more commonly used in mathematics is the rectangular coordinate system.  In this system, the velocity of fluid parcel can be “decomposed” or resolved into components along each of three coordinate axes, “x”, “y” and “z”, all of which are at right angles to one another. 

 

These axes are in the west-east, north-south, and up-down directions, respectively.   Since a fluid parcel can move in either direction on each of these axes, each axis has a positive and negative direction.  For the x-axis, motion from west to east is considered positive, for the y-axis, motion from the south to the north is considered positive, and for the z-axis motion upwards is considered positive, totally consistent with the definition of the rectangular coordinate system by mathematicians.

 

In the rectangular coordinate system, the horizontal wind has two components, one along the x and one along the y axis.  To keep these two components separate, we give them different symbols, u and v, as shown below.

 

 

In both the rectangular and the natural coordinate system the vertical axis is the “z” axis, and the vertical wind is given the symbol w.

 

 

 

 

The expressions above can be used to illustrate how the wind components can be calculated. We'll use the wind component along the x-axis, u, as an example. Just to remind you, a west wind is u>0, an east wind is u<0.

 

Figure 1 looks down on the x and y axis. Say a fluid parcel (an air parcel or an ocean parcel) moves from the origin to 10 meters east of the origin, and that takes 1 second. We can use the first expression above to calculate the fluid parcel's speed.

 

initeDifferenceExample

 

Figure 1: Example of wind component magnitude calculation using finite differencing

 

 

The general rule is to take the value of the dependent variable, in this case, x, at the point furthest along the path in the direction of evaluation, subtract the value at the initial location and then divide by the time elapsed. You'll note that this gives the common sense answer of 10 meters/second for the example above. The fact that the answer is positive tells us that the fluid parcel is moving from west to east; a west wind.

 

The Relationship Between Velocity and Acceleration

 

But, as we will see, acceleration is a key principle in understanding fluid motion. Acceleration is the rate of change of velocity.  In other words, acceleration is a measure of the degree to which a fluid parcel’s velocity CHANGES.  It’s important to note that the velocity is a quanitity that has both direction and magnitude;  it’s a vector.  So is acceleration.

 

Thus a fluid parcel experiences acceleration if its direction of motion changes, even if it speed does not.  The symbol for acceleration is often given as “a”.  But it’s important to remember that, for example, in the rectangular coordinate system, the acceleration can be  along any of the coordinate axes.  The acceleration along the vertical axis for example, would be:

 

 

where the subscript indicates that the acceleration component is along the z axis.