ball drop project

7/23/2019 Ball Drop Project

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Ball DropProjectMath 2120

Professor Jean

Gerardo Gonzalez

Spring 2013



How do we find the distancetraveled for a tennis ball when

it is dropped from a known

height?We can figure out the distance traveled

for a tennis ball when it is dropped from aknown height by using calculus and

analytical software.



Procedure:

Step 1: To determine the distancetraveled you first must use a hi tech digital

video camera with high frames rates per second. Record the ball drop from aknown height and let the ball drop until itstops bouncing.

Then you must upload the video on to ananalytical software. I used logger pro.



Step 2: On logger pro set your scale to a knownheight. This is known as a reference point. My

scale was 60inches. Then set your origin wherethe ball hits the floor.

Step 3:Then mark the tennis ball with a pointwhen the ball is released and when the balltouches the floor. Do this consecutively until theball stops bouncing.

The great thing about logger pro is that it allowsyou to mark points when the tennis ball isreleased and when it touches the floor. These

points are then transferred to a graph. With thisgraph you are able to recognize that asequence is occurring.



Here I put points on the tennis ball and I also useda cool feature that lets me measure the distant of

each successive fall.



Like I said the neat thing about logger pro is that I canput dots on my tennis ball from top to bottom and

logger pro automatically makes a graph of it. This is agraph that represents the tennis ball bouncing fromtop to bottom. If you are an amazing math whiz youwill recognize that this is a sequence.



Unfortunately, since my tennis ball goes

beneath the origin, I am not able to get the

last few measurements. If I would haveincluded that last few measurements I would

have gotten results that were inconclusive

because the tennis ball is beneath the origin.

The tennis ball also starts to move a bit to this

side. This causes lots of trouble for logger probecause logger pro has trouble measuring the

distance of the ball drop when the tennis ball

moves out of place. In brief, the angle at

which the video was shot, and the fact that

the ball moved out of its place causes logger pro to measure the distance of the ball drop

inaccurately.



After your done gathering your measurements with logger pro, you can continue to the calculus part of the project.

With the measurements and graph that were obtained by using

Logger Pro, we are able to develop a sequence.

A sequence is a set of ordered values. My values for my sequenceare{h } 59.58,33.69,18.49,10.69,6.163n



0

1

2

3

4

Initial Height:59.58inches

Values:

59.58

33.69

18.49

10.69

6.163

h

h

h

h

h

• The reason why I don’t have a lot of

measurements is because the last fewbounces were underneath the originon Logger Pro. Like I said earlier if Iwould have measured the last few ballbounces I would have came out withnegative values for the measurements.

• We have our measurements, but we are missing some importantdetails. These details are crucial to get the total distant traveled.

• Since I only measured the ball dropping from top to bottom, Ihave to take into consideration that the ball also traveled frombottom to top on each measurement except the first one.

• I will have to multiply the series that will be shown in the futureslides by 2. I then subtract 59.58 from the total distance traveledbecause I multiplied the first measurement by 2.



{h } 59.58,33.69,18.49,10.69,6.163n

As in all calculus books, a sequence can be related to aseries. Interestingly, every series has a related sequence ofpartial sums. We use r to figure out the pattern of thesequence of partial sums that is related to the series.

Original sequence: { } 59.58,33.69,18.49,10.69,6.163nh

Sequence of Partial Sums: { } 59.58,33.72,19.08,10.80,6.11nS



Finding a Pattern To find the sequence of partial sums we have to

determine a pattern from one height to the next.

With this pattern we are able to create asequence of partial sums.

To figure out the pattern, we have to divide thesecond height by the first height to see thepattern. This results in what we call a ratio or percentage. After we figure out the individual

ratios we add up all ratios and divide them by thetotal number of ratios. This is our avg. ratio.

1 0/h h



Finding out what R is.

The ratio changes as you move down thesequence.

When all the ratios are calculated, theavg ratio for my problem is

1 0

2 1

3 2

4 3

/ h 33.69 / 59.58 .565

/ 18.49 / 33.69 .548

/ 10.69 /18.49 .578

/ 6.163 /10.69 .576

:.566

h

h h

h h

h h

AvgRatio



Now that we have our avg.

ratio we can find our sequence of partial sums.Abbreviated:

0

1 0

2 0

3 0

4 0

59.58

(.566) 33.72

(.566)(.566) 19.08

(.566)(.566)(.566) 10.80

(.566)(.566)(.566)(.566) 6.11

h

h h

h h

h h

h h

nS



The pattern that was revealed:

0

0

( )

is the inital height

r is the ratiois the term in the sequence

n

nS h r

h

n



The sequence of partial sum issimilar to the geometric series.

However, the sequence of partial sumsdoes not contain a sum.

The geometric series does.

S

0

0

0

0

( ) Does not contain sum.

( ) This series does contain a sum and is slightly in the form of a partial sum,

however, with a few changes it results in a Geometric series.

The changes are =

n

n

n

k

S h r

h r

h

0

a and

k

k

n k

ar



Geometric Series

a is our initial height=59.58inches

r is the ratio: .566

k is the number in the sequence

0

k

k

ar



Computing the Infinite SeriesWe know that r has to be less than 1.

Because if it is not it diverges.

If r is less than on then that means that theGeometric series is convergent. By thegeometric series being convergent wecan use the formula

1

n

aS

r



•

I only included measurements from top to bottom, notfrom bottom to top in my sequences. That’s why I

multiplied the series by 2 so I can account for thebottom to top of my measurements.

• I also had to subtract 59.58 from the distant traveledbecause I multiplied by 2 to the initial measurement.

Results:

0 0

2 59.58(.566) = 119.16(.566)

119.16274.56 59.58 214.98

1 1 (.566)

k k

k k

ainches

r



Total Distant Traveled:214.98inches

The End!

ball drop project

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