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TExES Physics/Mathematics 7-12 (243): Practice & Study Guide62 chapters | 688 lessons | 60 flashcard sets

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.

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 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).

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** 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).

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.

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.

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.

We all know gravity as the force that holds us and other objects down. But what is it really? Gravity is a force that exists between any two objects of mass (essentially everything in the physical universe), the strength of which depends on the objects' masses and distance from each other. Strictly speaking, you are pulling on the Earth as much as it is pulling on you. However, since the Earth is so much more massive than you are, you would not notice the Earth being pulled up towards you when you do a pull-up. But we digress.

Gravity pulls all objects down at the same rate of acceleration, 9.8 m/s^2. If you dropped a bowling ball from a plane, after 1 second its velocity would be 9.8 m/s (downward). After two seconds, its velocity would be 19.6 m/s. After 3 seconds, its velocity would be 29.4 m/s, 4 seconds 39.2 m/s. However, because of the forces of air drag (and buoyancy, especially in water or other mediums thicker than air), objects will have a **terminal velocity,** a velocity beyond which further acceleration does not occur. For example, a human skydiver in a belly down position has a terminal velocity of approximately 55 m/s downwards. At terminal velocity, acceleration is zero, since its velocity is not changing over time. In fact, a bullet fired from an airplane will actually demonstrate *negative acceleration* as it travels down towards Earth, from an initial velocity of 1000 m/s to 100 m/s.

Speed is the same as velocity only when the direction of motion is a straight line. Even then, the two are not strictly the same - they just have the same number (e.g., 9 mph and 9 mph west). The rate of change for velocity is acceleration, which is measured in displacement over time over time (e.g., m / s^2). Most real-world examples of acceleration (like a sprinter) are not constant. However, acceleration due to gravity does occur at a constant rate, until buoyant and drag forces become significant as velocity increases.

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TExES Physics/Mathematics 7-12 (243): Practice & Study Guide62 chapters | 688 lessons | 60 flashcard sets

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