Activity on the average speed of two speeds

Exercise 1. RESET the applet. Set the Number of Bins to 20 and the Speed to 24.0 m/s. PLAY the motion until the car has completed one round, as closely as you can make it. Change the speed to 48.0 m/s, and continue the motion until the car has completed another round.

Calculate the car's average speed for the two rounds, and compare your result with the exact value for the average speed displayed on the Speed vs. Time graph or in the Data box. A slight difference between the two values must be expected because in the actual motion the car will not have gone exactly one round at 24 m/s and exactly one round at 48 m/s.

Exercise 2. Don't reset the applet. Display the Speed vs. Time graph for the previous motion, and observe that the average speed for the two rounds (horizontal blue line) is closer to 24 m/s than to 48 m/s.

Observe that the time interval during which the car travels at 24-m/s is longer than the time interval during which it travels at 48-m/s. In your Notebook, write a few sentences explaining how these two facts are related.

Exercise 3. Don't reset the applet.

It is true for any speed vs. time graph that the area under the graph for a time interval of duration Δt is equal to the distance s traveled during the time interval. If one indicates the average speed during this time interval by a horizontal line, as in the applet's Speed vs. Time graph, he distance traveled is also equal to the area under this line because s = v_avΔt.

Verify that the areas under the applet's Speed vs. Time graph and under the blue average-speed line, both taken for the entire two rounds, are equal.

You can approximately read the times elapsed during which the speeds of 24 m/s and 48 m/s were maintained from the graph. You can get the exact values from the Data box. Bins 9 and 19 contain the speeds 24 m/s and 48 m/s, respectively. The times in the column headed "Acc. Time: t(s)" are the exact times elapsed while these speeds were maintained during the simulated motion.

Exercise 4. Don't reset the applet. Press the Data button to display the table of speed bins. With the number of bins equal to 20, the speed interval from 0 to 50 m/s is subdivided into 20 bins of width 2.5 m/s. Find the bins containing the speeds 24.0 m/s and 48.0 m/s.

Verify the bin speed average listed below the bin table as follows. For each bin, take the speed v at the midpoint of each bin, the bin speed, and multiply it by the total time t during which the car's speed was in this bin (this is called the accumulated time). Add all products vt and divide by the total time elapsed (sum of the times accumulated in all bins).

The bin speed average is close to, but in general not equal to, the exact average speed because in general the bin speeds (taken at the midpoint of each bin) are not exactly equal to the true speeds.

Exercise 5. Do not reset the applet. Display the Accumulated Time vs. Speed graph together with the bin table (Data button). In a few sentences, explain how the graph represents the data in the bin table.

Exercise 6. RESET the applet. Set the number of bins to 20. PLAY the motion at a speed of 12.0 m/s for 10.0 s (or as close to 10.0 s as possible), PAUSE the motion, then PLAY it at 24.0 m/s for another 10.0 s (or as close to 10.0 s as possible). In your Notebook, make a graph of speed vs. time for this motion, calculate the average speed and indicate it by a horizontal line in your graph. Also make a graph of accumulated time vs. speed for this motion, using bins of 2.5 m/s width. Compare your graphs and calculations with the applet.

Exercise 7. Repeat Exercise 7 several times, but each time with a different duration of the 24.0-m/s speed interval. Choose 20.0 s, 30.0, and 40.0 s for this time interval. RESET the applet before each repeat.

Exercise 8. RESET the applet. Compare two motions: one in which a speed of 12.0 m/s is maintained for 10.0 s and followed by a speed of 24.0 m/s maintained for 20.0 s with another motion with a speed of 24.0 m/s maintained for for 10.0 s, followed by a speed of 12.0 m/s maintained for 10.0 s, followed by a speed of 24.0 m/s maintained for 10.0 s. Both motions have the same accumulated times for the 12.0 m/s and 24.0-m/s speeds, namely, 10.0 and 20.0 s, respectively.

Calculate and compare the average speeds for the two motions, and compare the Accumulated Time vs. Speed and Speed vs. Time graphs for the two motions. Also compare the data in the corresponding bin tables. In a few sentences, describe and explain what you find.

Activity on the average speed of three speeds

Exercise 1. The average speed v_av of two speeds v₁ and v₂, maintained during time intervals of duration t₁ and t₂, respectively, is equal to

v_av = (t₁ / t) v₁ + (t₂ / t) v₂(1)

where t = t₁ + t₂ is the total travel time.

For a derivation of Equ.(1) see Kinematics/Speed/Average/Explain It/Weighted Average, Page 3. Modifying the derivation appropriately, show that the average speed of three speeds v₁, v₂, and v₃, maintained during time intervals of duration t₁, t₂, and t₃, respectively, is equal to

v_av = (t₁ / t) v₁ + (t₂ / t) v₂ + (t₃ / t) v₃(2)

where t is again the total travel time.

Exercise 2. RESET the applet. Make the car move through three rounds, the first round at 10 m/s, the second round at 20 m/s, and the third round at 40 m/s. Use Equ.(2) to calculate the average speed, and compare your value with that given by the applet.

Exercise 3.