Position, Velocity, Acceleration

Position

Position(measured in meters or m) provides the spacial location an object or entity occupies in space. Position is often referred to as the variables s or d. 


Position is a vector quantity so it has magnitude and direction. This means that if an object is 5 meters away from another object, the magnitude of its position is 5 meters. However, its direction is defined by where it is in relation to the other object. The object could be 5 m above, below, to the right, or to the left of the other object. This is why keeping position as a vector is important. Intuitively, you can say that an object's instantaneous position is where it is in space at a given point in time.


Displacement is the change in an object's position, given by Δx(the final position - the initial position). Displacement, of course, also has a magnitude and direction making it a vector quantity. The magnitude of displacement is the distance(not the same thing), both of which are measured in meters(m).


To illustrate the difference between distance and displacement, let's take a hypothetical scenario where you're running around a circular track, as you would in track and field. The standard high school track has a circumference of 400 meters. If you run exactly one lap around the track, you'd run a distance of 400 meters. However, since your final position equals your initial position, your displacement would be 0 meters because your position didn't change.


Velocity

Velocity is defined as the rate of change of an object's position with respect to time. Intuitively, this is interpreted as "how does an object's position change as time goes on?". Velocity is also a vector quantity and its magnitude is speed(which has no direction), both of which are measured in meters/second(m/s).


​ If the position of an object is changing, the object must have some velocity, regardless of direction. When you start your car from rest, it goes from having no velocity(and speed) to having a non-zero velocity(and speed) in a given direction. Something important to remember is that if an object changes direction, its velocity will change even if it goes at the same speed as before. ​


Now that we've introduced velocity a bit, we can introduce some fundamental mechanical equations that relate position and velocity.


The equation above states that the change in position divided by the change in time gives the average velocity of an object. Now, given this, we can formulate a proper relationship between velocity and position. However, this equation is usually applied to the beginning and end of a given time interval. Let's try and apply to a more infinitesimal time interval. ​An infinitesimal time interval is another word for an extremely small time interval. This equation uses a derivative, which is a calculus term for the instantaneous slope of a function.


Taking the average velocity over a large time interval is equivalent to measuring someone's car speed based on a 2 hour road trip. The average wouldn't be a good measure of what the trip was like because the car could've been stopping, turning, hitting red lights, going on freeways, and more such that its velocity would change a lot and not really be close to the average at all. However, if we take a bunch of infinitesimal points along the road trip, finding the car's velocity at each tiny segment of the trip, and map it out, we have a much better idea of what exactly the trip was like. This is the idea of instantaneous vs average velocity, given by the equation above.

Acceleration

Acceleration is defined as the rate of change of velocity with respect to time. In terms of calculus, this could be said as "acceleration is the derivative of velocity with respect to time". Since velocity is measured in meters/second, acceleration will be measured in meters/(second squared). Acceleration can also be seen as the rate of change of the rate of change of the position of an object with respect to time. Looking at acceleration like this defines it as the second derivative of position with respect to time. Only an object at constant speed and direction has no acceleration, otherwise, if an object's speed and/or direction is changing, it's accelerating.


The calculus derivative form of acceleration is expressed as below:

Citations/Attributions