# Speed, Velocity & Acceleration

Instructor: Thomas Higginbotham

Tom has taught math / science at secondary & post-secondary, and a K-12 school administrator. He has a B.S. in Biology and a PhD in Curriculum & Instruction.

Speed and velocity are not the same thing. Objects' speed and velocity can be identical or very different, depending on their directions of motion. In this article, learn more about these foundational physics concepts, including acceleration and free-falling objects.

## Race Car Physics

Daytona 500 drivers race their cars 200 times around a track, for a total of 500 miles. Those cars go fast, right? Average speeds of over 160 mph. Joey Logano won the race in 2015. His average velocity during the race? Zero mph. Wait, what? To understand this requires a review of the physics definitions of speed and velocity, which sets us up to discuss acceleration.

## Speed vs. Velocity

Speed is a scalar quantity, one that requires only a magnitude (miles per hour, here). Velocity is a vector quantity, one that requires a magnitude (also mph) and a direction. How do these get measured?

Speed is a measure of distance traveled over a certain amount of time. In the Daytona 500, cars travel a distance of 500 miles in approximately 3.1 hours. To calculate speed, 500 miles / 3.1 hours = 161 miles/hour.

However, velocity is a different story since it measures displacement over time, as opposed to distance. Displacement is a measure of how far from the starting point an object ends up, similar to the colloquial, 'as the crow flies.'

In the case of the Daytona race, cars started and ended at the same point, despite having traveled a distance of 500 miles. To calculate velocity, 0 miles / 3.1 hours = 0 miles / hour (in any direction).

## Velocity Further Clarified

Let's assume the car that starts at the Pole Position (i.e., the first in line) begins right on the start line. If the start line is the same as the finish line, then their displacement for the entire race will always be 0. Let's also assume that a car in the rear of the pack starts 528 feet south of the start line. That car will actually have a displacement for the entire race of 528 feet north, or 0.1 mile. If that rear car takes 3.1 hours to finish, its velocity would be 0.1 miles north / 3.1 hours = 0.0002 miles north per hour. Cars that start in the rear of the pack are pretty much guaranteed to have a higher velocity than those in the front of the pack. This also illustrates real-world considerations vs. physics considerations. No racer would purposely start at the back of the pack just to end up with a higher velocity.

## Acceleration

Acceleration is also a vector quantity, requiring both a magnitude and a direction. Acceleration is often called 'speeding up,' though it would more accurately be referred to as 'velocitying up.' Still, to understand acceleration, it is probably easiest if we think about straight line acceleration, in which case speed and velocity are similar (since direction is not changed).

Let's pretend a Daytona car takes 4 seconds (0.0011 hours) to go from rest to a velocity of 60 miles per hour (north), at a constant rate of acceleration.

A = change in velocity / time

Change in velocity = final velocity / initial velocity.

In the current example, change in velocity is:

60 mph - 0 mph = 60 mph

Acceleration (north) = 60 mph / 4 seconds = 15 mph (north) / second. Note that the units are displacement / time / time. The SI unit for acceleration is meters per second per second (m/s^2).

## Negative Acceleration

Colloquially, we often hear the term 'deceleration' to describe slowing down, as in the case of a car that screeches to a halt to avoid a fender bender. In physics, there is no such term. Instead, physicists use the term 'negative acceleration' to describe a decrease in an object's velocity over time.

## Real-world Acceleration

Much to the chagrin of physics teachers everywhere, most real-world examples of acceleration do not occur at a constant rate. In the case of the race car, consider the car going from 0 to 40 mph in the first two seconds, and then from 40 to 60 mph in the next two seconds. During the first two seconds, the acceleration rate was 20 mph (north) / second, while for the remaining two seconds, the acceleration rate was 10 mph (north) / second. It is pretty uncommon for objects in our daily lives to have constant rate of acceleration.

## Acceleration Due to Gravity

Take a bowling ball and a golf ball to the fifth story of a building and drop them at the same time. Which hits first? Neither: they both hit the ground at the same time because gravity accelerates them downward at the same rate. In fact, when air resistance is negligible, all objects accelerate downward at the same rate due to gravity. If we were able to construct a tall vacuum (an area without any matter, including air), a bowling ball and a feather dropped at the same time would hit the ground at the same time.

Specifically, the rate of acceleration due to gravity is 9.8 m/s^2.

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