Introduction to Velocity


February 29, 31 and 7, 2001

Katherine Chaison
Lab Partner: Bruce Jackson


Abstract:

In this lab we used a photo gate in combination with a motion sensor to study instantaneous velocity, average velocity and the difference between the two types of velocity. Video analysis was also used in this lab to aid in the analysis.


Theory:

The parameters position, velocity and acceleration are most commonly used to describe one dimension motion. Velocity is the change of position over the change of time. Instantaneous velocity is the velocity of an object at a certain point in time whereas average velocity is the ratio of the total distance an object travels over a set time period and the set time period it takes the object to travel that certain distance. Velocity can be constant, positive, or negative. A constant velocity is a velocity with no acceleration. Velocity is positive when the object is moving forward with either positive or negative acceleration and velocity is negative when the object is moving backward with either positive or negative acceleration. Two different devices, the motion sensor and the photo gate, can be used to measure velocity. A motion sensor measures the position of objects at fixed time intervals and computes the velocity from the differences in the readings. The photo gate works the opposite way, measuring the time of an object when it reaches fixed distance intervals and calculates the velocity by the difference between the readings. Besides being measured by a motion sensor and photo gate, velocity can also be measured by taking the motion of an object and completing a video analysis.


Experiment:

  1. First we set up an experiment that measured velocity with both the motion sensor and the photo gate. We set up a horizontal track flat on the ground with the motion sensor at one end of the track and the photo gate positioned around the middle track. The cart was set up with the picket fence card containing the twelve black boxes at the top, lining up correctly with the photo gate. Then we placed the cart at the beginning of the track and gave it a push. This motion triggered the motion sensor to start recording and as soon as the cart passed the photo gate the photo gate started recording. We repeated the above steps making additional measurements with different initial velocities.
  2. Next we removed the motion sensor and measured the velocity of the cart using the photo gate only. We continued taking measurements with the photo gate using different initial velocities.
  3. After measuring the velocity of the cart using the photo gate and the motion sensor we took a break from that equipment and continued to study velocity using video analysis instead. We opened the Space_shuttle.vpt and the Lunar_Module.vpt files on our computer. These two files contained video clips of motion. First we watched the video clips several times. This allowed us to find a point of reference on each object. It is important to make sure that the chosen point of reference exists and remains unobstructed throughout the whole video. Once a point of reference had been established, we positioned the cross hairs on the point of reference for each frame of the film. This allowed the computer to calculate the velocity and acceleration of the object. This process was carried out on both video files.
  4. Finally we studied average velocity as a function of the distance between two photo gates in experiment PO3. We set up the track with one of the ends raised 21 cm off the table, providing an inclination for the cart to travel down. We put the first photo gate in the center of the track which was 60 cm. Next we used the "Velocity - Intro" setup to measure the instantaneous velocity of the cart using the 12-bands on top of the five-band picket fence, making sure that the photo gate was aligned with the correct band. We repeated the measurement of the instantaneous velocity of the cart two additional times, each time setting the cart on the top of the track and releasing it. So each measurement had the same initial velocity. After measuring the instantaneous velocity we attached another photo gate to the track and positioned both photo gates 40 cm away from the midpoint of 60 cm, with one photo gate secured at 20 cm, and the other at 100 cm. We flipped the card on the cart so that it was using the five-pattern picket fence on top and adjusted the photo gates so that they were both aligned to read the right band. Once both photo gates were attached to the correct positions we released the cart down the track three different times, taking measurements each time. After three reads had been taken we moved the photo gates to 30 cm from the midpoint, establishing one at 30 cm and 90 cm. We took three more readings and then moved the photo gate to 20 cm and then 10 cm away from the midpoint, repeating the steps and measurements that were carried out at the previous two positions. It is important to note that the above measurements have all been carried out with the same angle of inclination of the track. After data had been obtained for 40 cm, 30 cm, 20 cm, and 10 cm away from the midpoint we changed the height of the elevated end of the track to 26 cm. With the changed angle of elevation we carried out the same steps and measurements that we used for the previous angle of elevation on the new angle of elevation, including the measurement of the instantaneous velocity of the cart using "Velocity- Intro". Once all the readings and procedures had been carried out for the second angle of elevation we raised the elevated end of the track to 30 cm, and repeated the whole entire process of the two previous angle of elevations again.


Data Analysis:

  1. In the data collected from the first experiment where we were measuring velocity with both the motion sensor and the photogate and comparing the results of the two different techniques. Both the motion sensor and the photogate measure the instantaneous velocity of the cart at specific intervals and display that instantaneous velocity on the graph. From the three different data runs, it was possible to determine that each way of measuring velocity was consistent in the way that it measured it. Since the velocity for each run was achieved the same way with the only difference being the force behind each push, the graphs for each different run should show the same trend, with different starting velocities. For the most part, the graphs are consistent with each other when the motion sensor measured the velocity. Figures 1,2 and 3 show the graphs of velocity measured with the motion sensor for runs 1,2 and 3. Note the similarities in the curves of the graph. For velocity measured with a motion sensor the initial velocity is high with a sudden drop and then a relative constant velocity of velocity. Upon closer inspection of figures 1 and 2 small dips of the curve below zero are noticeable. A possible explanation of these dips is that something interfered with the motion detector.


Figure 1. Velocity graph with a motion sensor for Run 1

Figure 2. Velocity graph with motion sensor for Run 2.

Figure 3. Velocity graph with motion sensor for Run 3.

However, when velocity is measured by the photogate, the graphs do not show the same trend. Instead, three different graphs are produced with no similar characteristics. The differences in the graphs could be due to the way that the photogate measures velocity. Figure 5 shows the graph for the largest initial velocity. If one notes in the graph there is a constant velocity for about .06 sec. It is possible that at such as high velocity the cart whizzed right through the photogate, barely slowing down, explain the constant velocity on the graph. Figure 4 is the velocity graph of the medium initial velocity. The graph of the velocity shows a downward trend regardless of the spikes that appear in the graph. Figure 6, which is the velocity graph of the lowest initial velocity, demonstrates a curve which is similar to those seen when measuring velocity with a motion sensor.

 


Figure 4. Velocity graph with photogate for Run 1.


Figure 5. Velocity graph with photogate for Run 2.

Figure 6. Velocity graph with photogate for Run 3.

Because both the motion sensor and the photogate are measuring the same motion, it would be expected that the data from both the motion sensor and photogate would correspond with each other. Figures 7 and 8 are the velocity-velocity graphs for runs 1 and 3. The velocity-velocity graph for each run plots the velocity measured by the motion sensor against the velocity measured by the photogate.


Figure 7. Velocity vs. Velocity for Run 1


Figure 8. Velocity vs. Velocity for Run 2

While the majority of the points on each velocity vs. velocity graph do not correspond with each other exactly, many points are only slightly off, proving that the motion sensor and photogate measure the same velocity.
  1. From the data that was collected from using just the photogate we were able to determine that the graphs of the smaller velocities demonstrated the same trend. For the larger velocity, after initially following the same trend we found that there was deviation from the other two graphs. At first glance it appears that the velocity for each graph was constant. After zooming in on the velocity graph though it becomes apparent that that is not the case. Figures 9, 10, and 11 are the velocity graphs of each run with a different starting velocity, with figure 9 having the smallest initial velocity, figure ten the medium initial velocity and figure eleven the greatest initial velocity. For figures nine and ten, the velocity graph starts out with a small increase in velocity and then a progressive downward trend in the graph as the velocity grows smaller. This can be explained by the idea that by pushing the cart to give it its' initial velocity, for a short while the velocity increases because of the momentum behind the force. Then the cart starts to slow down and move with a lower velocity because it is no longer feeling the effects of the initial push and the friction between the track and the cart has begun to slow it down. Figure eleven is a harder velocity graph to explain. In the beginning it shows the same trend as the other two graphs, the increase in the velocity of the cart, however when the cart starts to slow down, the velocity will decrease and then suddenly increase. This could possibly be explained by the fact that because the cart was started at a greater initial velocity, it was harder for the photogate to get an accurate read or it could have missed a picket fence block or it could be explained by human error. However it is important to note that in figure eleven the difference between each velocity point is in the thousandths while in the other two figures the difference is in the hundredths.



    Figure 9. Run 1's Velocity Graph, Smallest Initial Velocity


    Figure 10. Run 2's Velocity Graph

Figure 11. Run 3's Velocity Graph, Largest Initial Velocity

  1. Using the video clips of motion we were able to watch motion in slow motion as with each video still the object moved. Because we knew the time increment between each still, it was possible to figure out the instantaneous and average velocity of the object. By placing the crosshairs on a point of the object that remained visible for all the stills, plotting the placement of each crosshair produced the velocity graph.
  1. In experiment P03, average velocity was measured using two photogates and the difference between the photogates. Tables 1, 2 and 3 show the basic data for each run done in P03.

Height 21 cm
Trial #'s
Distance between photogates (cm)
1-3
40
4-6
30
7-9
20
10-12
10

Table 1. Distance data for Run 1

Height 26 cm
Trial #'s
Distance between photogates (cm)
1-3
40
4-6
30
7-9
20
10-12
10

Table 2. Distance data for Run 2

Height 30 cm
Trial #'s
Distance between photogates (cm)
1-3
40
4-6
30
7-9
20
10-12
10

Table 3. Distance data for Run 3


For each different distance between the two photogates we took three different measurements to ensure accuracy. Table 4 shows the data for each different distance as well as the margin of error for each distance for run 1 as a sample of the margin of error encountered on each run.

Trial #
Distance between photogates (cm)
Average Speed (m/sec)
Margin of Error
1
40
.5820
+/-.0005
2
40
.5835
3
40
.5830
4
30
.575
+/-.0023
5
30
.574
6
30
.568
7
20
.5510
+/-.00083
8
20
.54950
9
20
.54850
10
10
.5105
+/-.0003
11
10
.5080
12
10
.5080

Table 4. Distance and Margin of Error data for Run 1

 

Figure 12 displays the graphs for runs 1, 2 and 3 for the average speed of experiment P03. If one was using the graph of the average speed in trying to determine the instantaneous speed, the measurement that would most accurately reflect the instantaneous speed on the graph would be the data points to the farthest right of the graph, or the data points taken with the least distance between the photogates. The graph of the instantaneous velocity for runs 1, 2 and 3 is displayed by Figure 13 for comparison to the average speed graphs of Figure 12.


Figure 12. Average Speed Graphs for Runs 1, 2 and 3

Figure 13. Velocity Graphs Corresponding to Heights Used in P03

Conclusion:

Through our experimentation with velocity, using two different ways of collecting data, both the motion sensor and the photogate, we were able to explore average, instantaneous and constant velocity. With our first experiment comparing the results of the motion sensor and the photogate we discovered that even though each instrument had separate ways of collecting data, both were consistent in the way they collected and displayed data. While the data from the motion sensor and the photogate did not correlate exactly, the data was not so far away each other that we determined that both the photogate and motion sensor were accurately able to detect velocity. Next using just the photogate, we discovered that the velocity graph of an object that at first appeared constant upon closer inspection was not, enforcing the idea that constant velocity is when the velocity of an object remains constant, meaning the velocity is neither increasing or decreasing. Measurement of velocity besides taken by the motion sensor or photogate was also explored when we used the stills from a video clip to determine the motion of a lunar module and a space shuttle. Since we knew the time increment between each still, by determining the height of the object in each still, it was easy to find the velocity graph. Finally in experiment P03 the relationship between instantaneous velocity and average velocity was explored. It was determined that when plugging in an x value on the average velocity graph that the corresponding y value does not produce the accurate instantaneous velocity for that value of x. Instead the y value of the average velocity graph is more like a rough estimate for the instantaneous velocity. However, the more data points that make up the average velocity graph, the closer the y value is to being the correct instantaneous value. Average velocity is normally calculated by taking the distance over the time taken to travel that distance. No such simple formula exists for finding instantaneous velocity, but devices such as a motion sensor or photogate can measure instantaneous velocity.


Remarks:

Learning about the differences between average and instantaneous velocity was fun. However, the experiments could have been accomplished a lot less painstakingly and in less time by my partner and I if we had been aware of the photogate cord interfering with the cart earlier.