Velocity

We use vectors not only to describe the position of an object but also to describe velocity (speed and direction). If we know an object's present speed in meters per second and the object's direction of motion, we can predict where it will be a short time into the future. As we have seen, change of velocity is an indication of interaction. We need to be able to work with velocities of objects in 3D, so we need to learn how to use 3D vectors to represent velocities. After learning how to describe velocity in 3D, we will also learn how to describe change of velocity, which is related to interactions.

The concept of speed is a familiar one. Speed is a single number, so it is a scalar quantity (speed is the magnitude of velocity). A world-class sprinter can run 100 meters in 10 seconds. We say the sprinter's average speed is (100 m)/(10 s) = 10 m/s. In SI units speed is measured in meters per second, abbreviated “m/s.”

A car that travels 100 miles in 2 hours has an average speed of (100 mi)/(2 hr) = 50 miles per hour (about 22 m/s). In symbols,

where v_avg is the “average speed, ” d is the distance the car has traveled, and t is the elapsed time.

There are other useful versions of the basic relationship among average speed, distance, and time. For example,

expresses the fact that if you run 5 m/s for 7 seconds you go 35 meters. Or you can use

to calculate that to go 3000 miles in an airplane that flies at 600 miles per hour will take 5 hours.

While it is easy to make a mistake in one of the formulas relating speed, time interval, and change in position, it is also easy to catch such a mistake by looking at the units. If you had written

, you would discover that the right-hand side has units of (m/s)/m, or 1/s, not s. Always check units!

If a car went 70 miles per hour for the first hour and 30 miles an hour for the second hour, it would still go 100 miles in 2 hours, with an average speed of 50 miles per hour. Note that during this 2-hour interval, the car was almost never actually traveling at its average speed of 50 miles per hour.

To find the “instantaneous” speed—the speed of the car at a particular instant—we should observe the short distance the car goes in a very short time, such as a hundredth of a second: If the car moves 0.3 meters in 0.01 s, its instantaneous speed is 30 meters per second.

Earlier we calculated vector differences between two different objects. The vector difference

represented a relative position vector—the position of object 2 relative to object 1 at a particular time. Now we will be concerned with the change of position of one object during a time interval, and

will represent the “displacement” of this single object during the time interval, where

is the initial 3D position and

is the final 3D position (note that as with relative position vectors, we always calculate “final minus initial”). Dividing the (vector) displacement by the (scalar) time interval t_f − t_i (final time minus initial time) gives the average (vector) velocity of the object:

Definition: Average Velocity

Another way of writing this expression, using the “Δ” symbol (Greek capital delta, defined in Section 1.5) to represent a change in a quantity, is

Remember that this is a compact notation for

Consider a bee in flight (Figure 1.31). At time t_i = 15 s after 9:00 AM, the bee's position vector was

. At time t_f = 15.1 s after 9:00 AM, the bee's position vector was

. On the diagram, we draw and label the vectors

and

Figure 1.31

The displacement vector points from initial position to final position.

Next, on the diagram, we draw and label the vector

, with the tail of the vector at the bee's initial position. One useful way to think about this graphically is to ask yourself what vector needs to be added to the initial vector

to make the final vector

, since

can be written in the form

The vector we just drew, the change in position

, is called the “displacement” of the bee during this time interval. This displacement vector points from the initial position to the final position, and we always calculate displacement as “final minus initial.”

Note that the displacement

refers to the positions of one object (the bee) at two different times, not the position of one object relative to a second object at one particular time. However, the vector subtraction is the same kind of operation for either kind of situation.

We calculate the bee's displacement vector numerically by taking the difference of the two vectors, final minus initial:

This numerical result should be consistent with our graphical construction. Look at the components of

in Figure 1.31. Do you see that this vector has an x component of +1 and a y component of −0.5 m? Note that the (vector) displacement

is in the direction of the bee's motion.

The average velocity of the bee, a vector quantity, is the (vector) displacement

divided by the (scalar) time interval t_f − t_i. Calculate the bee's average velocity:

Since we divided

by a scalar (t_f − t_i), the average velocity

points in the direction of the bee's motion, if the bee flew in a straight line.

What is the speed of the bee?

What is the direction of the bee's motion, expressed as a unit vector?

Note that the “m/s” units cancel; the result is dimensionless. We can check that this really is a unit vector:

This is not quite 1.0 due to rounding the velocity coordinates and speed to three significant figures.

We can plot the average velocity vector on the same graph that we use for showing the vector positions of the bee (Figure 1.32). However, note that velocity has units of meters per second whereas positions have units of meters, so we're mixing apples and oranges.

Figure 1.32

Average velocity vector: displacement divided by time interval.

Moreover, the magnitude of the vector, 11.18 m/s, doesn't fit on a graph that is only 5 units wide (in meters). It is standard practice in such situations to scale down the arrow representing the vector to fit on the graph, preserving the correct direction. In Figure 1.32 we've scaled down the velocity vector by about a factor of 3 to make the arrow fit on the graph. Of course if there is more than one velocity vector we use the same scale factor for all the velocity vectors. The same kind of scaling is used with other physical quantities that are vectors, such as force and momentum, which we will encounter later.

We can rewrite the velocity relationship in the form

That is, the (vector) displacement of an object is its average (vector) velocity times the time interval. This is just the vector version of the simple notion that if you run at a speed of 7 m/s for 5 s you move a distance of (7 m/s)(5 s) = 35 m, or that a car going 50 miles per hour for 2 hours goes (50 mi/hr)(2 hr) = 100 miles.

a valid vector relation? Yes, multiplying a vector

times a scalar (t_f − t_i) yields a vector. We make a further rearrangement to obtain a relation for updating the position when we know the velocity:

The Position Update Formula

This equation says that if we know the starting position, the average velocity, and the time interval, we can predict the final position. This equation will be important throughout our work.

The position update formula

is a vector equation, so we can write out its full component form:

Because the x component on the left of the equation must equal the x component on the right (and similarly for the y and z components), this compact vector equation represents three separate component equations:

example


	Updating the Position of a Ball At time t_i = 12.18 s after 1:30 PM a ball's position vector is . The ball's velocity at that moment is . At time t_f = 12.21 s after 1:30 PM, where is the ball, assuming that its velocity hardly changes during this short time interval?

Solution

Note that if the velocity changes significantly during the time interval, in either magnitude or direction, our prediction for the new position may not be very accurate. In this case the velocity at the initial time could differ significantly from the average velocity during the time interval.

The curved colored line in Figure 1.33 shows the path of a ball through the air. The colored dots mark the ball's position at time intervals of one second. While the ball is in the air, its velocity is constantly changing, due to interactions with the Earth (gravity) and with the air (air resistance).

Figure 1.33

The trajectory of a ball through air. The axes represent the x and y distance from the ball's initial location; each square on the grid corresponds to 10 meters. Three different displacements, corresponding to three different time intervals, are indicated by arrows on the diagram.

Suppose we ask: What is the velocity of the ball at the precise instant that it reaches location B? This quantity would be called the “instantaneous velocity” of the ball. We can start by approximating the instantaneous velocity of the ball by finding its average velocity over some larger time interval.

The table in Figure 1.34 shows the time and the position of the ball for each location marked by a colored dot in Figure 1.33. We can use these data to calculate the average velocity of the ball over three different intervals, by finding the ball's displacement during each interval, and dividing by the appropriate Δ t for that interval:

Figure 1.34

Table showing elapsed time and position of the ball at each location marked by a dot in Figure 1.33.

Not surprisingly, the average velocities over these different time intervals are not the same, because both the direction of the ball's motion and the speed of the ball were changing continuously during its flight. The three average velocity vectors that we calculated are shown in Figure 1.35.

question

	Which of the three average velocity vectors depicted in Figure 1.35 best approximates the instantaneous velocity of the ball at location B?

Figure 1.35

The three different average velocity vectors calculated above are shown by three arrows, each with its tail at location B. Note that since the units of velocity are m/s, these arrows use a different scale from the distance scale used for the path of the ball. The three arrows representing average velocities are drawn with their tails at the location of interest. The dashed arrow represents the actual instantaneous velocity of the ball at location B.

Simply by looking at the diagram, we can tell that

is closest to the actual instantaneous velocity of the ball at location B, because its direction is closest to the direction in which the ball is actually traveling. Because the direction of the instantaneous velocity is the direction in which the ball is moving at a particular instant, the instantaneous velocity is tangent to the ball's path. Of the three average velocity vectors we calculated,

best approximates a tangent to the path of the ball. Evidently

, the velocity calculated with the shortest time interval, t_C − t_B, is the best approximation to the instantaneous velocity at location B. If we used even smaller values of Δ t in our calculation of average velocity, such as 0.1 second, or 0.01 second, or 0.001 second, we would presumably have better and better estimates of the actual instantaneous velocity of the object at the instant when it passes location B.

Two important ideas have emerged from this discussion:

The direction of the instantaneous velocity of an object is tangent to the path of the object's motion.

Smaller time intervals yield more accurate estimates of instantaneous velocity.

You may already have learned about derivatives in calculus. The instantaneous velocity is a derivative, the limit of

as the time interval Δ t used in the calculation gets closer and closer to zero:

In Figure 1.35, the process of taking the limit is illustrated graphically. As smaller values of Δ t are used in the calculation, the average velocity vectors approach the limiting value: the actual instantaneous velocity.

The rate of change of a vector (the derivative) is itself a vector.

A useful way to see the meaning of the derivative of a vector is to consider the components:

The derivative of the position vector

gives components that are the components of the velocity, as we should expect.

Informally, you can think of

as a very small (“infinitesimal”) displacement, and dt as a very small (“infinitesimal”) time interval. It is as though we had continued the process illustrated in Figure 1.35 to smaller and smaller time intervals, down to an extremely tiny time interval dt with a correspondingly tiny displacement

. The ratio of these tiny quantities is the instantaneous velocity.

The ratio of these two tiny quantities need not be small. For example, suppose that an object moves in the x direction a tiny distance of 1 × 10⁻¹⁵ m, the radius of a proton, in a very short time interval of 1 × 10⁻²³ s:

which is one-third the speed of light (3 × 10⁸ m/s)!

Velocity is the time rate of change of position:

. Similarly, we define “acceleration” as the time rate of change of velocity:

. Acceleration, which is itself a vector quantity, has units of meters per second per second, written as m/s/s or m/s².

Definition: Acceleration

Instantaneous acceleration is the time rate of change of velocity:

Average acceleration can be calculated from a change in velocity:

The units of acceleration are m/s².

If a car traveling in a straight line speeds up from 20 m/s to 26 m/s in 3 seconds, we say that the magnitude of the acceleration is (26 − 20)/3 = 2 m/s/s. If you drop a rock, its speed increases 9.8 m/s every second, so its acceleration is 9.8 m/s/s, as long as air resistance is negligible.

There are two parts to the acceleration, the time rate of change of the velocity

As we'll see in later chapters, these two parts of the acceleration are associated with pushing or pulling parallel to the motion (changing the speed) or perpendicular to the motion (changing the direction).