position, velocity, and acceleration

In the last module, you learned about vectors and vector addition. This will come in handy when we start working with two-dimensional projectile motion in the next lesson. For now, just keep in mind that the following concepts are all vectors:

Position or displacement is the distance between an object's starting and ending locations (meters).

Velocity is the speed and direction of an object's motion. This is also the rate of change of an object's position (meters/second).

Acceleration is how quickly something is speeding up or slowing down. This is also the rate of change of an object's velocity (meters/second^2).

A few facts to dispel some common misconceptions:

Distance and displacement are not the same thing. For example, think about a typical physics class. I pace back and forth at a ridiculous rate sometimes. I travel many, many meters as I do this, but I end up at roughly the same place I started. This means my distance traveled is large, but my displacement is close to zero.
- - You can calculate displacement as Δx = xf – x0. Here, "x" represents horizontal position in meters. In physics, Δ always means some sort of change. Simply put, this equation reads as "final position minus initial position."
A high acceleration does NOT mean the object is traveling quickly. It simply means the object's speed is changing at a rapid pace. In fact, it's entirely possible to have a high acceleration and a velocity of zero m/s at the same moment in time!
A high velocity does NOT mean the object has a high acceleration. In fact, objects traveling at constant high speeds have an acceleration of zero m/s^2.
It is certainly possible to have both high velocity and high acceleration, but the presence of one does not indicate the other.

On the previous page, I defined velocity as being the rate of change of position. This would lead a lot of people to define an equation for velocity as being:

Velocity = Displacement / Time

Written another way,

**Special note: for this equation above, and for several other equations this unit, you'll see little subscript x's. This is to indicate that these equations hold only in one direction at a time. The average horizontal velocity is the change in horizontal position over time. If I wanted to find the average vertical velocity, I could change all the x's to y's.**

This is a solid definition for average velocity, but it doesn't take into account the act of speeding up or slowing down.

For example, think of a 100-meter dash. Athletes line up, begin at rest (0 m/s), then speed up as they run. If you divided 100 meters by the time they took, you'd find their average speed across the entire race, but in reality, they began at a lower velocity and ended at a higher velocity.

Instantaneous velocity is how fast something is moving at any given moment in time.

If I instead made a spreadsheet and found an athlete's speed at every second of the race, I'd be getting closer to the idea of instantaneous velocity. If I made one to discover the speed at every half second of the race, I'd be getting even closer. Instantaneous velocity tells me how fast something is moving at a single...well...instant. This is often more useful than calculating average velocities for situations where an object's speed is constantly changing, such as projectile motion, automobile accident investigations, and more.

To properly find an object's instantaneous velocity, a little bit of Calculus is required. Rather than simply being the total change in an object's displacement over the total time of travel, take this definition and compress it into as small a change in displacement over as small a change in time as you can imagine. A change so small, it's infinitesimally close to 0. That's right, folks - an object's velocity is a derivative of its position function over time:

acceleration

Average acceleration is defined as the change in an object's velocity over time:

Guess what! To find the instantaneous acceleration, you'd take the derivative of velocity!

Note that acceleration refers to any change in an object's velocity. This can include speeding up, slowing down, or changing direction.

By the way, stating that acceleration is the derivative of velocity means it's also the derivative of the DERIVATIVE of position! This is what's called a second derivative. Here's the way second derivatives can be written:

Image:
Sivayula. (n.d.). [Photograph]. Retrieved from http://www.everythingmaths.co.za/science/grade-10/21-motion-in-one-dimension/21-motion-in-one-dimension-05.cnxmlplusVanderburg, Steven. (n.d.). Lightning [Photograph]. Retrieved from https://phys.org/news/2016-09-instantaneous.html Zajda, Tomasz. (n.d.). Velocity [Digital Image]. Retrieved from https://www.britannica.com/story/whats-the-difference-between-speed-and-velocity

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