1.3 Velocity and Speed

Key concepts: 

• Velocity = displacement / time

• Speed = distance / time

• Position, displacement, and velocity are some of the variables of motion.

• Equations of motion tie together variables of motion and allow us to model the motion using math.

Important side-quests:

• Vectors are measurements that must have a direction. Vector notation communicates magnitude and direction.

• Rates of change measure the change of one quantity compared with the change of another quantity.

Textbook sections:  2.3

Velocity is an important physics concept, and it is NOT the just a fancy word for speed. Velocity is a vector, which means that it must have some sort of direction associated with it. We have already met two vectors: position and displacement. This is the third one, and it’s worth taking a moment to review.

What, exactly, is a vector? (Side quest)

A vector is any measurement that MUST have a direction associated with it in order to make sense. For example, thinking back to our definition of position from module 1.2, in order to know the position of an object we need to know two things:

• How far is the object from the zero point? (This is referred to as the “magnitude” of the vector.)

• Which direction is the object from the zero point? (This is referred to as the “direction” — which is pretty obvious!)

Let’s zoom in on the concepts of magnitude and direction:

• Magnitude: This is an actual measurement. It has dimensions (length, or time, or maybe some combination of them) and it has units (meters, or seconds, or some other units of length and time). So, you might see a position vector of x = +3.75 m. The 3.75 m part is the magnitude.

• Direction: This is where it gets a little confusing, because there are lots of options. Here they are:

  • Algebraic sign: Remember when we first started this journey that we defined a frame of reference as a set of coordinates that we would imagine existing over the stuff we would measure. A set of coordinates includes an origin — the zero point — and one positive direction per axis. So, looking at our example above, the position is 3.75 meters from the zero point along the positive x axis.

  • Words that mean positive or negative: We (in the Western world) often think of “right” as the positive direction, which is probably because we read from left to right. We also think of “up” as a positive direction. Thus, “left” and “down” usually mean negative direction. Usually. There is no law that says “up and right must be positive.” It’s just a convention that we use a lot.

In one dimension, vectors are usually just written as signed numbers. The sign is the direction and the number is the magnitude.

Vectors in two dimensions may be written in several ways, and we will explore this more fully in Unit 2. For now, here’s the key idea to keep in mind: A two dimensional vector will be expressed as two signed numbers. You can extend the idea: a three-dimensional vector will be written as three signed numbers. In general, you need one number per dimension.

Scalars are measurements that do not require a direction to make sense. Mass is a scalar quantity. To put it another way, scalars are the ordinary one-number measurements that you have been using all along.

Review ended: On to the new stuff

Velocity:

The key idea of velocity is that it is displacement over time. Displacement. Not distance. So remember that displacement is:

We can measure a time interval in the same way:

Putting those two together, we get a mathematical definition of velocity:

Velocity has dimensions of [distance] / [time]. The normal metric unit is meters per second, written as m/s. However, any distance over any time is a potential velocity unit, so km/h is something you will see as well.

Velocity is a vector because displacement is a vector. Displacement is a vector because position is a vector. See the chain? In general, if you add or subtract or (in some cases) multiply a vector, you get another vector as the result. So going back to our vector lesson above, velocity will usually be written as a signed number if we’re in one dimension, two signed numbers if we’re in two dimensions, and so on.

Rates of Change (Side quest)

Rate of change is a concept that we often see in physics (and calculus). The idea is that two things are changing at once. In the case of velocity, position is changing and so is time. So the key question is:

How fast does one thing change compared to the other?

Velocity is an answer to that question. For every second that goes by, some number of meters (maybe positive, maybe negative) also goes by. This is called a “rate of change with respect to time” because we started by saying “for every second that goes by…” There are other rates of change, as you will see later in the course!

Speed:

If we take the idea of velocity — the change of position over time — and we replace change of position with the distance traveled, we get speed. It’s not just a word change: there are important differences.

• Speed: distance over time. The path matters! Think back to Module 1.2 and the American football analogy. The ball has a displacement that is equal to the yardage gain. However, the distance that the ball travels is often much larger than its displacement. Since distance is a scalar, not a vector, the speed is always positive.

• Velocity: displacement over time. Only the endpoints matter. We do not care how the ball got from the line of scrimmage to its final destination. We only care about the yardage gain (+) or yardage loss (-).

If the motion of the object is in a straight line and in the positive direction, the velocity and speed will be the same number. For any other situation, they will probably be different.

Variables of Motion and Equations of Motion:

“Variables of Motion” is really just a catch-all name for the concepts that we have been talking about. So far, we have learned four variables of motion:

• Time: We use the letter t for time and we usually measure time in seconds.

• Position: We use x and y as letters for position, and we usually measure them in meters.

• Displacement: We use Δx and Δy for displacement. We measure displacement in meters as well.

• Velocity: We use v for velocity, and sometimes we put an x or y subscript after the v to indicate that it is an x or y velocity. We measure velocity in meters per second.

“Equations of motion” is the name for mathematical equations that use the variables of motion. For example, the equation for velocity:

… is an equation of motion. Equations of motion are just like any other equations. They can be solved for variables, but they can also be used to understand or describe a situation. For example, if we take the velocity equation above and graph it, we get something that looks like this:

Read the axis labels carefully. The position axis is vertical, so the x variable is actually on the “y-axis.” (This is why your algebra teacher probably told you not to get hung up on calling them the x and y axes!) The time axis is the horizontal. So, from our equation above, the change in position is actually the “rise” and the change in time is the “run”. That means that the velocity is the slope!

Just about every algebra student has learned the slope-intercept equation of a line:

The same equation, but with the variables changed to match the graph is a new equation of motion that we derived from the earlier equation of motion.

Equations of motion serve two important purposes in physics:

• Predicting: If we know an initial position and a velocity, we can answer the question “where will I find the object when the time is… whatever the time is.

• Understanding: When we have a mathematical description of the motion, we can answer questions about why the object might be doing what it’s doing. This is going to be important in Unit 3. For now, just know that it’s coming!

If you made it this far, even if it took a couple days to get here, congratulations. This was a lot of information to digest, and it will take quite a lot of practice to become truly familiar with these ideas and the many different ways that they will appear.

To Do:

• Try some problems: 5, 9, 11, 13

• Try this at home