The daily ramblings …

Geophysics : An Introduction to Velocity

Posted in Geophysics by akamal on January 18, 2007

[ Introduction]

Velocity by definition refers to the rate at which an object changes its position. It is of a vector quantity, and as such, is direction aware. To illustrate this, imagine a person moving rapidly, one step forward and another backward, always returning to the original position. While this might result in a change of activity, it would result in a zero velocity, as the person always return to the original position (no distance to comapare with).

In geophysics, velocity is an extremely important trait that enables us to predict the sort of content the earth’s ground covers. As it is related to how fast waves travel between layers of the earth, it also tells us of how compact they are in between.

It’s formula is:

velocity = total distance / total time , or put simply,
v = d/t

As in the usual case of physics, there are 2 types of fundamental velocity to deal with. Let’s have a quick look into all 2.

[ Average Velocity ]
Much like the name suggests, it actually calculates the average speed between the distance traveled and time taken. For example, when traveling to Petaling which is 50km in distance, and taking 1 hour, my velocity would be

v = 50 km / 1 hour = 50 km/h

It’s comparison will be your starting distance and time against your last.

[ Instantaneous Velocity ]
Shows the velocity at one given point. As the name indicated, it’s a velocity at one instance of time. In average velocity, we consider the speed between point A and B, so they’re like intervals between them.

[ Example ]

Consider the example below (don’t bother to ask where I got the diagram, I freakingly did it myself) :

Velocity Diagram 001

Consider a car going from point A to B. As the diagram indicated, the car arrives at point B in 50 mintues time. To calculate the average speed, we use the following information.

total distance = sum of distance; td = 25 + 20 + 30 + 10 = 85 km
total time = end time – starting time; tt = 50 – 0 = 50 minutes

The average velocity is:

td / tt = 85
50 = 1.7 km per minute

In geophysics, the standard is meter/second, so I’d take the answer as 24 meter/second.

Now, to understand the concept of instantaneous velocity, I’d have to write some lengthy explanation here. Remember that instantaneous refers to a reading at one particular point or time.

Suppose the velocity of the car is varying, because for example, you’re in a traffic jam. You look at the speedometer and it’s varying a lot, all the way from zero to 60 mph. What is the instantaneous velocity? It is, more or less, what you read on the speedometer. I’m assuming you’ve got a good speedometer that isn’t too sluggish and can change its reading quite quickly. Your speedometer is measuring the the average velocity but one measured over quite a short time, to ensure that you’re getting an up to date reading of your velocity.

So if you measure the displacement of the car d over t, you can use that to determine the average velocity of the car. What we want is to take the limit as difference of t goes to zero. More formally, the instantaneous velocity v is defined as

v = d / t; where t -> is nearing to 0

What the heck do I mean by that? I know you’d be confused. Just consider this, instant velocity still uses the same equation as of average. v = d/t, but instead of taking the total difference between the 2 variables, we try to limit the range of the difference, example, to get velocity at 40km, we use these information:

time difference = 160 seconds – 159.5 seconds
distance difference = 40km – 39.75km

The comparison looks so small right? That’s what you call instantaneous velocity, because we’re comparing the velocity at time 160 seconds, as compared to the previous velocity at 159.5 seconds. Usually, with average, you’re just comparing from time at some xx seconds and 0 (initial time). So what is the limit you say? Yeah, there is a limit to what extent you should define the difference on. You can’t just write difference of time as 60 seconds – 59.999999 seconds, that’s absurd.

To explain on the concept of limits, I’ll explain it in another article.