Using Time and Displacement to Calculate the Velocity of a Ball Rolling Down a 41.8 cm Board

Sami S. Wright

 

Introduction

Calculating an object’s average velocity is important because it is applicable to real-life situations, such as how fast a car is going over a period of time. In this experiment, the amount of time it takes for a ball to roll down a wooden board and the length of the board will be measured. These measurements will then be used to find the average velocity of the ball as it rolls down the board at different time intervals. Average velocity is found when an object’s velocity is measured over an extended period of time (Wile, 13). The purpose of this experiment is to calculate an object’s velocity. If velocity is the object’s displacement over a period of time, then average velocity can be calculated if position of the ball rolling down the board is measured over multiple time intervals.

 

Materials

• iPhone stopwatch clock app
• A plastic centimeter ruler (30 cm long).
• A pile of thick chapter books, 7 cm high.
• The back of a wooden puzzle board (41.8 cm long).
• A mechanical pencil.
• A mesh/rubber ball.

Procedure

1. Gather the materials.
2. Make a rough estimate of the center of the board. Draw a line all the way across.
3. Place one end of the board on top of the stack of books to form an incline.
4. Using the centimeter ruler, measure the distance from the top of the board to the mark made in the middle. Label this distance “d₁“.
5. Measure the distance from the mark to the bottom of the board and label this distance, “d₂“.
6. After taking the measurements and ensuring that the board is stable, ready the stopwatch on the iPhone clock app and hold the ball at the top of the board.
7. Release the ball and simultaneously start the stopwatch.
8. When the ball reaches the halfway mark, pause the stopwatch. Record the time as shown on the iPhone.
9. Repeat step 8 four more times. When all five results are recorded, average the numbers and record the result.
10. Take the average found in step 9 and divide “d₁” by this number. The answer is the velocity. Label this “v₁“.
11. Repeat steps 6-9. This time, however, when releasing the ball, do not start the stopwatch until the ball rolls across the halfway mark. Stop the watch when the ball reaches the end of the board.
12. Take the average calculated in step 9, after repeating the steps, and divide “d₂” by this number to find the velocity. Label the measurement “v₂“.
13. Repeat steps 6-9 once more. This time, start the stopwatch when the ball is released at the top of the board, and stop the watch when the ball reaches the bottom of the board.
14. Take the average calculated in step 9, after repeating the steps, and divide “d₁” and “d₂” combined by this number. Label the result “v₃“.
15. Clean up the materials.

 

Results

Board measurements:

• “d₁” = 22.2 cm
• “d₂” = 19.6 cm

Table 1. Shows the times, averages, and velocities measured during this experiment.

Round #1 (Start stopwatch when ball is released at top of board and stop when ball reaches halfway mark.)	Round #2 (Start stopwatch when ball reaches halfway mark and stop when ball reaches the end of the board.)	Round #3 (Start stopwatch when ball is released at the top of board and stop when ball reaches the end of the board.)
#1: 0.73 sec	#1: 0.48 sec	#1: 1.23 sec
#2: 0.78 sec	#2: 0.38 sec	#2: 1.04 sec
#3: 0.78 sec	#3: 0.43 sec	#3: 0.91 sec
#4: 0.64 sec	#4: 0.40 sec	#4: 1.13 sec
#5: 0.78 sec	#5: 0.56 sec	#5: 1.07 sec
Average = 0.74 sec	Average = 0.45 sec	Average = 1.08 sec
“v1” = 30.0 m/sec	“v2” = 43.6 m/sec	“v3” = 38.7 m/sec

 

Equation 1: v = d/t

This equation shows that by dividing the measurements from the board by the average times of the ball, the velocity was able to be calculated.

The results show the amounts of time it took the ball to roll down different sections of the board and the averages of each round’s times. The velocities were found using the board’s measurements for displacement and the average time in each round calculated together using the velocity equation. The results helped to explain how average velocity works and also presented us with an example of instantaneous velocity.

 

Discussion

The results from this experiment show that “v₁” is less than “v₂” and “v₃” is right in between them. Why did this happen? Well, as the ball travelled down the board, it continued to gain speed. So “v₁” is the lowest because the ball had not gained a lot of speed by the time it rolled to the halfway mark. “V₂” was much greater than “v₁” because the ball had picked up more speed before the clock even started. Finally, “v₃” was in between “v₁” and “v₂” because it was the average velocity of the other two measurements.

 

A brief example of instantaneous velocity is also given through the results of this experiment. Instantaneous velocity is an object’s velocity when it is measured at one moment in time (Wile, 13). Considering the results of this experiment, “v₁” was the average velocity of the ball when it travelled down the first half board. “V₂” was the average velocity of the ball when it travelled down the second half of the board, and “v₃” was the average velocity of the ball when it travelled down the entire length of the board. Because “v₁” and “v₂” were measured over shorter periods of time than “v₃“, “v₁” and “v2″ are closer to instantaneous velocities than “v₃” (Wile, 15).

 

The results of this experiment are important because they can help determine critical information in real life situations. For example, it is very important for an Olympic runner or sprinter to know his/her average velocity so they can know what they need to improve on during their training to be able to win their races.

An error that can occur while performing this experiment is, while calculating the averages, forgetting to divide by the number of numbers. If this happens, the answers will be very different and will not make sense.

 

The results of this experiment demonstrate the definition of average velocity by calculating “the velocity of an object over an extended period of time.” After recording  how much time it took the ball to roll down the board during the different rounds and finding each velocity using those numbers combined, the ball’s velocity over an extended period of time was found. Therefore, the velocity of the ball was not found in just one roll, but it took us a total of fifteen rolls to find the velocity of each round. Average velocity was successfully calculated using the position of the ball rolling down the board over multiple time intervals.

 

References

Wile, Jay. Exploring Creation with Physics. Anderson, IN: Apologia Educational Ministries, 2004. Print.