Josh has taught Earth Science and Physical Science at the High School level and holds a Master of Education degree from UNC-Chapel Hill.
When driving, never take your eyes off the road! That said, it's necessary to occasionally glance at the speedometer. When you do, you are reading your instantaneous velocity. This lesson explores the science behind instantaneous velocity.
Definition of Instantaneous Velocity
You worked hard all week and it's finally here: the weekend. You and some friends decide to take a short trip to your favorite spot just a few hours' drive from where you live (for me, it's the beach). You're cruising along the freeway, listening to music without a care in the world. But you start to care when you see a parked police vehicle on the side of the road. You start to care a lot. You frantically glance at your speedometer and see that you are a few miles under the limit. You continue to cruise, recovering from your momentary panic.
The reading you took from the car's speedometer is instantaneous, meaning it represents only a single moment in time. If you were to watch the speedometer, take notes on the readings, and average them, you would have an average reading. For our purposes in this lesson, we'll focus on instantaneous measurements.
To most people, the word velocity means how fast something is traveling. Those people are not wrong. Velocity does, indeed, refer to the quickness or slowness of something, and by that simplified definition, instantaneous velocity is the velocity reading (how fast or slow you're going) at a particular point in time.
This may be the common definition of velocity applied in everyday speech, but scientists would disagree slightly. They would say that this simplified definition is actually the definition of speed, and that there is actually slight scientific difference between velocity and speed. Let's explore this further.
The Difference Between Velocity and Speed
Let's start with speed. Speed is defined as the rate of distance of travel over time. Simply put, this means that speed is the distance you travel divided by the time it took to travel that distance. So, it is how fast or slow you are moving. Speed is all about this change in distance over an amount of time; it has nothing to do with the direction you're traveling.
Velocity, on the other hand, takes into account the direction an object is traveling. It still is concerned with the speed, but the direction must also be reported. An example of a speed is '45 miles per hour.' An example of velocity is '45 miles per hour west.' In the real world, people often ignore the importance of including direction in a velocity measurement, so the words speed and velocity sometimes become interchangeable. If you tell someone your instantaneous velocity when unfortunately passing a parked police vehicle was 90 miles per hour, they probably wouldn't prompt you for the direction.
A Better Definition
Using what we now know about the difference between velocity and speed, we can better define instantaneous velocity as the rate of change in the position of an object over time. Let's focus on the part of that definition that says, 'the position of an object.' It's easiest to think about position as if you're referring to a map or coordinate plane. On both of these reference systems, a change in position measures both distance and direction. Since velocity includes the direction traveled, this definition is much more complete and accurate than the one many people use interchangeably with the definition of speed.
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For our purposes, we're going to focus simply on the speed portion of the instantaneous velocity measurement and leave the direction out. The direction can usually be included rather easily, since it's generally a direct measurement. The formula for instantaneous velocity is depicted in the on-screen now.
Formula for velocity
Let's break down the formula for velocity. In this formula, v represents velocity, x represents the position of the object, and t represents time. The triangle-shaped symbol is called delta, and means, 'the change in.' So, deltax means, 'the change in position.' Let's do an example using this formula.
A car is traveling southeast and travels 40 miles in 30 minutes. What is the instantaneous velocity, assuming the car is traveling the same speed the entire time, in miles per hour halfway through the trip?
Breaking down the question, it does not really matter which point in time we look at, but let's look at the halfway point, since that is what it asks for specifically. Halfway through the trip, the car has traveled 20 miles in 15 minutes. We want this to be reported in miles per hour, but we have the time being given in minutes. It's easy to fix that, though. Just divide the number of minutes given by the number of minutes in an hour. In this case, 15 / 60 = .25 hours. Now, we just plug in our numbers to the original formula. 20 / .25 = 80 miles per hour southeast. Notice that I included the direction, which is part of velocity. Looks like this person was going pretty fast! I'm guessing they would have prompted our parked police officer to chase them down.
Anything that is instantaneous means that a reading is taken at a particular instant in time, while velocity is essentially how fast something or someone is travelling. Speed is the distance traveled over time, so instantaneous speed would simply be the distance traveled over time at a particular point in time, and if the speed is uniform throughout the example, the instantaneous speed wouldn't change. Instantaneous velocity is very similar to instantaneous speed, but it has the added factor of direction traveled. For a quick recap, here's the formula for velocity:
Formula for velocity
Remember, v represents velocity, x represents the position of the object, and t represents time. The triangle-shaped symbol is called delta and means 'the change in,' so delta x equals the change in position. Now you should have the tools available for calculating instantaneous velocity. Just remember to add the direction the object is going.
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