Drop Ball Lab

Following the ball rolling down the ramp lab and the cart ramp lab, there may still be questions about how the mass of the object effects its acceleration. The primary objective is to look at the relationship between the mass of an object and the acceleration of gravity. The second objective is to look at a quadratic model d = (1/2) a t^2 +v_it + d_i or in vertex form d = (1/2) a (t - h)^2 + k.

This page will walk you through the collection and demonstrate how to analyze the data.

Materials

1 motion detector and CBL or a CBR (Ranger)
1 TI-8? calculator

Software

Using the DROPBALL or DROPCHOO programs which are available as drop.83g to collect the data for a distance vs time graph of an object dropped. DROPBALL will provide a complete graph of the data while DROPCHOO selects a certain portion of the data and shifts the data so that linearization will work appropriately. If you use DROPBALL, the program CHOOSE can be used to select out only the portion of the data which is to be used in the lab evaluation.

Data Collection

Each pair of students is given a ball with a different mass.

Position the motion detector or ranger a few feet over the persons head. Another student can even hold it, although the results may vary. Run the program DROPBALL or DROPCHOO. The instructions will first ask to measure the distance to the floor to establish a baseline. All person should be away from the motion detector. Next, position the ball at least 0.5 meters below the motion detector. The person running the program should press ENTER to begin sampling data. When the individual hears the motion detector clicking, the ball should be dropped. Accurate results can be obtained even if the ball is dropped a second after the sampling begins.

The individual dropping the ball needs to stand away from the motion detector, move their arms sideways away from the detector and step back from the detector after dropping. The individual above left their hands in the range of the detector resulting in erroneous data being collected.
	Every once in a while, we even stop to have a good time or laugh. Above, Mr. Rynearson and two students intently prepare to release the ball. On a serious note, it is very important to move the arms away and down as the ball is released to record the motion of the ball and not the hands.
A sample data collection is shown above. The next step will be to Choose out the portion which is the ball falling. If using Dropchoo, this will be the next portion of the program. If using Dropball, run the program CHOOSE. With either program, use the cursor keys to select the portion at the left and right sides of the region to analyze. It is not imperative mathematically to get the first point where it is dropped. In fact, it is better to wait a few points after the drop to analyze as the ball may be moved in the hand of the individual releasing it.
	When the data has been selected, a graph similar to the one above will appear. Note: DROPCHOO overwrites the data and stores this graph in L1 & L2. DROPBALL and CHOOSE leaves the original graph in L1 & L2 while storing the selected portion in L3 & L4.

Analyzing the Individual Data

The student can then find a quadratic regression y = ax^2 + bx + c on the appropriate lists using the calculator or Graphical Analysis. (L1 & L2 for DROPCHOO and L3 & L4 for DROPBALL/CHOOSE). The data from DROPCHOO may also be linearized successfully.

With the quadratic regression, recall that "a" is 1/2 of the acceleration, "b" is the initial velocity at t = 0 sec and "c" is the height off of the floor at t = 0 sec.

	The quadratic regression above shows that the acceleration is 2*-4.795 or -9.59 m/s^2. The accleration is negative because distance measured up from the floor is positive. The inital velocity is 2.30 m/s and an initial height at t=0 sec of 1.74 m. How can this be? Recall that the model assumes constant acceleration, so at time zero, the ball would have been travelling up toward the motion detector (positive velocity).
At the peak/vertex, the ball begins to fall. We only recorded data from the falling portion of the curve. Repeat the lab if you wish and throw the ball up towards the detector and allow it to fall. Note that the curve fits the line shown in the graph below. If you follow the curve back to the left, the height will begin to decrease as you reach t=0.
Mathematically, the vertex can be found by completing the square. From this method, we learn that the vertex is form is y=a(x-h)^2+k where h=-b/(2a) and k= c - b^2/(4a). The vertex of the parabola is the point at which the velocity is zero, or in this case the point in time where the ball was dropped.
In this case, the vertex is at (0.239, 2.01) indicating that the ball was drop 0.24 seconds after the motion detector started recording data and from a height of 2.01 meters above the floor.

Analyzing the Group Data

Recall that the purpose of the lab was to study if the mass of the ball had any significance in the acceleration of the ball when it was dropped. Each group was asked to record the mass of the ball and its acceleration on the board. The image below shows the results.

Questions for students:

1. Find a mathematical model for your "dropball" data.
2. Find the vertex of the mathematical model.
3. What do the values of "a", "b", "c", "h" and "k" mean for this lab activity? What does each value represent?
4. Find the class average mass of all the balls. Find the percent difference your ball was from the average mass.
5. Find the class average acceleration. Find the percent difference your ball was from the class average.
6. Does the mass of the ball have any significance in the acceleration? Justify your answer.

If you are an educator or interested in this activity and wish to have more information, send an e-mail to jrynear@lps.org or jwelker@lps.org.