Prerequisites

Students should understand the concepts of displacement and distance traveled and should have a working knowledge of vectors.

Learning Outcomes

Students will develop an understanding of the physics concepts of average velocity and average speed, will get more experience in working with vector and scalar quantities, and will learn about averages over time.

Instructions

The applet should be open. The step-by-step instructions in this lesson are to be carried out in the applet. You may need to toggle back and forth between instructions and applet if your screen space is limited. You should be prepared to do calculations to verify the numbers that are generated by the applet.

• Notation, Terminology, Definitions
• Average Velocity vs. Average Speed
• A Mistake to Avoid

1. _av, v_av, and |_av|.

   Average velocity is a vector and therefore denoted _av in boldface with an arrow on top. Average speed is a scalar and therefore denoted v_av in plainface without an arrow. When writing these symbols by hand, the only distinction between them is the arrow. Make sure you use it when you mean velocity.

   We will see below that the magnitude of the average velocity is in general not equal to the average speed. Therefore, one must use a different notation to distinguish between the two. The symbol |_av| will be used to denote the magnitude (absolute value) of the average velocity while v_av denotes average speed. This is an example where the magnitude of a vector cannot be denoted by the same symbol as the vector but without arrow.

2. , Δ, ||, d, and s.

   Displacement is a vector quantity denoted in boldface with an arrow on top. In this case, no confusion is possible if the magnitude of the displacement is denoted d. However, it could also be denoted ||. Another notation for displacement is Δ. It reflects the fact that displacement is the difference between (change in) two positions. Note that the magnitude d of a displacement is in general different from the distance s traveled. This is the reason why the letter d is not used in MAP to denote distance traveled.

3. Δt and t.

   Time elapsed measures an interval of time and is therefore properly denoted Δt, where the Delta symbol indicates a difference: Δt = t₂ - t₁, the differene between a later instant of time t₂ and an earlier instant of time t₁.

   E.g., the time elapsed between the two instants t₁ = 10 s and t₂ = 17 s is Δt = t₂ - t₁ = 17 - 10 = 7 s.

   The symbol t is used to denote an instant of time.

   If one considers a time interval extending from t = 0 to some later instant t of time, the time elapsed is Δt = t - 0 = t. In such a case, t can therefore be used to denote time elapsed. This is what is done in the applet.

4. Definition of average velocity and average speed.

   Average velocity is defined as the ratio: displacement divided by time elapsed. In symbols:

   _av = /Δt = Δ/Δt. (1)

5. Note that here we are dividing a displacement, not a "change" in a displacement, by the corresponding time elapsed. A displacement is a change in position, but a "change in displacement" does not make any sense here.

   Average speed is defined as the ratio: distance traveled divided by time elapsed. In symbols:

   v_av = s/Δt. (2)

   Again, you will find that in the applet Δt is replaced by t in these definitions, for the reason mentioned above.

6. Polar vs. Cartesian

   The applet gives values for vector quantities in two forms that will be referred to as polar and Cartesian in the following.

   In the polar form, a vector is specified in terms of its magnitude and direction, the latter specified by an angle relative to the positive x-axis in the positive (counter-clockwise) sense.

   In the Cartesian form, a vector is specified in terms of its x and y components.

Exercise 1. With the initial position of the ball set at (x,y)_i = (10.0,-15.0) m, drag the ball to near (x,y)_f = (-20.0,-10.0) m along a curved path, as in the snapshot in Figure 1 below. You may find it helpful to display the grid while setting the initial position and moving the ball.

Display the initial and final position vectors, _i and _f, respectively, and the data. Figure 1 below is an illustration of the kind of thing you should see. The frames around the data are omitted in Figure 1.

Figure 1

The following data are included among those in shown in Figure 1:

• distance traveled: s = 36.5 m
• displacement (polar form): = (d, θ) = (30.4 m, 170.5^o)
• displacement (Cartesian form): = (Δx, Δy) = (-30.0, 5.0) m
• time elapsed: t = 4.7 s

Question 1. Given these data, what is the average speed?

Answer. Using Definition (2) above gives

v_av = s/t = 36.5/4.7 = 7.8 m/s. (3)

This agrees, within round-off error, with the applet value displayed in Figure 1.

Question 2. Given these data, what is the average velocity? Give the answer in both the polar and Cartesian forms.

Answer (in polar form). Using Definition (1) above gives

_av = /t = (d/t, θ) = (30.4/4.7, 170.5^o) = (6.5 m/s, 170.5^o). (4)

This agrees, within round-off error, with the applet value displayed in Figure 1.

Note that to divide a vector by a scalar, one divides its magnitude by the scalar, but does not change the angle.

Observation. The magnitude of the average velocity has the value |_av| = 6.5 m/s, and the average speed has the value v_av = 7.8 m/s. The two values are not equal! This is the reason, pointed out at the beginning of the lesson, why the magnitude of the average velocity cannot be denoted by the symbol v_av. Why is the average speed greater than the magnitude of the average velocity? Could there ever be a situation in which it is the other way around?

Answer (in Cartesian form). Again, using Definition (1) gives

_av = /t = (Δx/t, Δy/t) = (-30.0/4.7, 5.0/4.7) = (-6.4, 1.1) m/s. (5)

This agrees, within round-off error, with the applet value displayed in Figure 1.

Note that to divide a vector by a scalar, one divides its x and y components separately by the scalar.

Exercise 2. Transform result (4) into form (5) and vice versa.

Exercise 1. Move the ball through two successive displacements ₁ and ₂, as in Figure 2 below, and determine the average speeds v_av,1 and v_av,2 and average velocities and _av,1 and _av,2 while these displacements take place.

Figure 2

Let the intermediate point, reached after the first displacement, be labeled "C". See Figure 2.

Data related to the intermediate point C are not displayed by the applet once the second displacement has been carried out. You need to observe the time elapsed during the motion from the initial point to point C at the end of the first displacement. For the motion shown in Figure 2 this time elapsed is:

time elapsed from the initial point to point C:

t_C = 2.5 s. (6)

Question 1. What are the values of the average speed during the first and second displacements for the motion in Figure 2?

Answer. The calculations are analogous to the one leading to result (3). For the first displacement, using for the distance traveled the value |x_C - x_i| = |x_f - x_i| = |-13.5 - 10.0| = 23.5 m, and using value (6), we get

v_av,1 = |x_C - x_i|/t_C = 23.5/2.5 = 9.4 m/s. (7)

For the second displacement, using for the distance traveled the value |y_f - y_C| = |y_f - y_i| = |9.6 - (-12.5)| = 22.1 m, and taking for the time elapsed t - t_C = 8.0 - 2.5 = 5.5 s, we get

v_av,2 = |y_f - y_C|/(t - t_C) = 22.1/5.5 = 4.0 m/s. (8)

Question 2. What are the values of the average velocity during the first and second displacements of the motion in Figure 2?

Answer. The calculations are based on Definition (1), and are analogous to calculation (5). It is an exercise for you to verify that the answers, in (x,y)-components, are

_av,1 = (-9.4, 0) m/s (9)

and

_av,2 = (0, 4.0) m/s (10)

Exercise 2. Determine the average speed v_av and average velocity _av for the entire movement created by you, combining both individual displacements.

Question 3. What is the average speed for the entire movement, combining both individual displacements, for the motion in Figure 2?

Answer. It may be tempting to work out the answer by taking the arithmetic mean of the two individual average speeds (7) and (8). This would give the value

(v_av,1 + v_av,2) = (9.4 + 4.0)/2 = 6.7 m/s. (11)

However, this value is incorrect because average speed is not defined as a mean of two average speeds over shorter time intervals. The correct value as shown in Figure 2 is

v_av = 5.7 m/s. (12)

To obtain value (12), use Definition (2) for the average speed. The total distance traveled is s = |x_C - x_i| + |y_f - y_C| = 23.5 + 22.1 = 45.6 m. Dividing this by the total time elapsed, 8.0 s, gives result (12).

Question 4. What is the average velocity for the entire movement, combining both the first and second displacements, for the motion in Figure 2?

Answer. Again, the correct answer is not equal to the mean of the two values (9) and (10). Work out this mean, and verify that it is not equal to the correct value for the average velocity _av = (-2.9, 2.8) m/s, shown in Figure 2.

Verify that this latter value is correct by using Definition (1), i.e., by dividing the total displacement, = (-23.5, 22.1) m, by the total time elapsed, t = 8.0 s.