Physics Lab

Lab Report Unit 2: FREE FALL & REACTION TIME 1

Lab Report Unit 2

Free Fall and Reaction Time

Diego Abad

Broward College

PHY2048L, Physics Cal I Lab

Dr. Delonia Wiggins

February 16th, 2021

Lab Report Unit 2: FREE FALL & REACTION TIME 2

I did the measurements of a free-falling tennis ball position from a y-coordinate image, and I

measured the displacement of a falling meter stick. Then I proceeded to calculate the value of the

displacement of the tennis ball position from mouse units into meters, to then calculate the time it

took for the ball to fall, and then calculate the g constant and find its percent error. Also, I

calculated time for the meter stick to fall, accounting for both air and no air, and then

calculating the percent error from an internet reaction time. The result was a measured value of

t =0.593s, g = 9.805, and percent error = 0.05%, and the meter stick data was t = 0.166s with a

percent error of 22.8%. I conclude that the experimental g constant it’s really close to the real

value of g, and that the reaction time between stopping a car by stepping the break will be

greater than my reaction time found in this experiment.

Introduction:

This lab will calculate the value displacement of a tennis ball falling behind two-meter

sticks, recorded in pictures from a camera that recorded the fall. After finding the average

displacement, we will calculate the constant g by finding the time it took for the fall to be as it is.

Then we will calculate the percentage error of the experimental value of the constant g with the

actual value of the constant g (9.8 . For the second part of this lab, we will measure the time of

reaction that takes to catch a falling meter stick, by first finding the average displacement that takes

to catch the meter stick. Then, we will use two formulas to calculate the time (one that does not

account for air resistance and one that does). Finally, we will use an average reaction time of

humans found on the internet to find the percentage error between the experimental value of time

and the internet value of time.

General Principles:

Lab Report Unit 2: FREE FALL & REACTION TIME 3

The first information we need to know to start finding the time of the fall of the Tennis ball

is how to convert the mouse units into meters. To do this, we will use a 2-meter constant since the

contains two-meter stick. Then, we will only have to divide 2 meters by the amount of mouse units

in between the 0-meter mark and the 2-meter mark of the ball in mouse units (displacement).

𝐷𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑌 𝑌

2.0𝑚 𝐷𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡

Next, we will need to know how to find the displacement in mouse units between the initial

position of the ball with its last positions and we will need to know the formula to calculate the

mean of anything:

𝑫𝒊𝒔𝒑𝒍𝒂𝒄𝒆𝒎𝒆𝒏𝒕 𝒀𝒇𝒊𝒏𝒂𝒍 𝒀𝒊𝒏𝒊𝒕𝒊𝒂𝒍

𝑴 𝑵

𝑴𝒊

𝑵

𝒊

Also, we will need to know how to convert mouse units into meters (which we will calculate when

we get the value of the displacement of the ball).

𝑿 𝒎𝒐𝒖𝒔𝒆 𝒖𝒏𝒊𝒕𝒔 ∗ . 𝟎𝒎

𝑫𝒊𝒔𝒑𝒍𝒂𝒄𝒆𝒎𝒆𝒏𝒕 𝒎𝒐𝒖𝒔𝒆 𝒖𝒏𝒊𝒕𝒔 𝑿𝒎

X= Displacement in mouse units

After finding the average displacement in meters, we will find the time by using this formula

bellow (we will solve for t in this case) and using 9.8 :

𝒚 𝒈𝒕 Æ 𝐲 𝐠𝒕 Æ 𝒕 𝒚 𝒈

Æ 𝒕 𝒚 𝒈

Lab Report Unit 2: FREE FALL & REACTION TIME 4

After finding t, we can plug in it in the same equation but this time solving it for g:

𝒚 𝒈𝒕 Æ 𝐲 𝐠𝒕 Æ 𝒈 𝒚 𝒕

Finally (for Part 1), we will need to use the formula to find the percentage error of an experimental

value and the expected value:

𝑷𝒆𝒓𝒄𝒆𝒏𝒕 𝑬𝒓𝒓𝒐𝒓 |𝑬𝒙𝒑𝒆𝒓𝒊𝒎𝒆𝒏𝒕𝒂𝒍 𝒗𝒂𝒍𝒖𝒆 𝑬𝒙𝒑𝒆𝒄𝒕𝒆𝒅 𝒗𝒂𝒍𝒖𝒆|

𝑬𝒙𝒑𝒆𝒄𝒕𝒆𝒅 𝒗𝒂𝒍𝒖𝒆 ∗ 𝟎𝟎

For Part 2, we will use the previous formulas from Part 1, the formula to find the time with air

resistance, and we will be using the formula for the uncertainty, the standard deviation, and the

relative error from the Lab 1 Manual:

Formula to find time with air resistance:

𝑻 𝒚 𝒈

𝒌𝒚 𝒎

Where y = displacement, g = 9.8, and k = 3.46*10-3 𝒌𝒈 𝒔

Formula to find Relative error steps bellow:

𝒅𝒊 𝑴𝒊 𝑴

Where 𝒅𝒊is the uncertainty of the measurement 𝑴𝒊.

𝑶𝒊 𝑵 |𝒅𝒊|

𝑵

𝑰

Lab Report Unit 2: FREE FALL & REACTION TIME 5

Where 𝑶𝒊 is the standard deviation, and ∑ |𝒅𝒊|𝑵𝑰 it is the sum of all squared deviations from each

measurement.

𝜀 𝑶𝒊 𝑴

Where 𝜀 is the relative error.

Methods:

For Part 1, I will start by measuring the finding the position of the two-meter mark in the

y-coordinate plane, and then the position of the zero-meter mark, to then find the difference

between both and use the formula of finding the rate of meters per mouse unit. Then, I will

calculate 5 times the initial and final position of the ball, find the difference of all 5 trials by using

the difference formula, and then I will convert it from mouse units into meters by using the mouse

unit per meter rate value that I will find before this step. After finding the difference in meters of

all 5 trials, I will calculate their individual times by using the t-solved formula without air

resistance, to then find the average of all 5 times. After that I will use the g-solved formula without

air resistance to find my experimental g value. Finally, I will calculate the percent error by using

the percent error formula in respect to the actual value of g (9.8 .

For part 2, I will calculate the displacement of a meter stick (from its 50cm mark) that is

freefalling in between one of my hands. To do this, I will have a partner hold the meter stick from

where I can see the 50cm mark inside my hand, and then I’ll wait for him to give the signal and

let the meter stick fall, graving it and then recording the final mark, and I will repeat this procedure

five times. After this, I will calculate the displacement by subtracting the final mark by 50cm, and

then I will convert this measurement in meters, to then start calculating its mean with the mean

Lab Report Unit 2: FREE FALL & REACTION TIME 6

formula, and its relative error by calculating its uncertainty, its standard deviation, and then using

the relative error formula. After that, I will calculate the average time using both formulas to find

t (with and without air resistance), and if both measurements result to be really like each other

(after rounding), I’ll use the round value of time. If not, I will use the average of both times. After

that, I will get the average reaction time of a human in the internet, to then calculate the percent

error of my experimental reaction time with the internet reaction time. Finally, I’ll answer if I

believe that the reaction time of stepping in a car break will be higher, lower, or equal to the

reaction time of this experiment.

Results:

Part 1

1.

a. Meter Stick at 2-meter mark (𝒎𝒐𝒖𝒔𝒆 𝒖𝒏𝒊𝒕𝒔) = 55

b. Meter Stick at 0-meter mark (𝒎𝒐𝒖𝒔𝒆 𝒖𝒏𝒊𝒕𝒔) = 500

Difference of measurements a and b (𝒎𝒐𝒖𝒔𝒆 𝒖𝒏𝒊𝒕𝒔) = 500 – 55 = 445

𝒎Meter per Mouse = 2.0m / 445 mouse units = 0.00449 𝟎. 𝟎𝟎 𝒎𝒐𝒖𝒔𝒆 𝒖𝒏𝒊𝒕𝒔

2.

Graph of Initial Y-Position and Final Y-Position of ball measured five times.

Trial # First Position (mouse units) Last Position (mouse units) 1 75 456 2 76 459 3 74 458 4 75 457

Lab Report Unit 2: FREE FALL & REACTION TIME 7

5 77 462

3.

Trial #1 Difference Y-Position: 456 – 75 = 381 𝒎𝒐𝒖𝒔𝒆 𝒖𝒏𝒊𝒕𝒔

Trial#2 Difference Y-Position: 459 – 76 = 383 𝒎𝒐𝒖𝒔𝒆 𝒖𝒏𝒊𝒕𝒔

Trial #3 Difference Y-Position: 458 – 74 = 384 𝒎𝒐𝒖𝒔𝒆 𝒖𝒏𝒊𝒕𝒔

Trial #4 Difference Y-Position: 457 – 75 = 382 𝒎𝒐𝒖𝒔𝒆 𝒖𝒏𝒊𝒕𝒔

Trial #5 Difference Y-Position: 462 – 77 = 385 𝒎𝒐𝒖𝒔𝒆 𝒖𝒏𝒊𝒕

4.

Trial #1 Difference in meters = 381 𝒎𝒐𝒖𝒔𝒆 𝒖𝒏𝒊𝒕𝒔*𝟎. 𝟎𝟎 𝒎 𝒎𝒐𝒖𝒔𝒆 𝒖𝒏𝒊𝒕𝒔

= 1.7145 1.715m

Trial # 2 Difference in meters = 383 𝒎𝒐𝒖𝒔𝒆 𝒖𝒏𝒊𝒕𝒔*𝟎. 𝟎𝟎 𝒎 𝒎𝒐𝒖𝒔𝒆 𝒖𝒏𝒊𝒕𝒔

= 1.7235 1.724m

Trial # 3 Difference in meters = 384 𝒎𝒐𝒖𝒔𝒆 𝒖𝒏𝒊𝒕𝒔*𝟎. 𝟎𝟎 𝒎 𝒎𝒐𝒖𝒔𝒆 𝒖𝒏𝒊𝒕𝒔

= 1.728m

Trial # 4 Difference in meters = 382 𝒎𝒐𝒖𝒔𝒆 𝒖𝒏𝒊𝒕𝒔*𝟎. 𝟎𝟎 𝒎 𝒎𝒐𝒖𝒔𝒆 𝒖𝒏𝒊𝒕𝒔

= 1.719m

Trial # 5 Difference in meters = 385 𝒎𝒐𝒖𝒔𝒆 𝒖𝒏𝒊𝒕𝒔*𝟎. 𝟎𝟎 𝒎 𝒎𝒐𝒖𝒔𝒆 𝒖𝒏𝒊𝒕𝒔

= 1.7325 1.733m

5.

Finding T:

Time #1 = ∗ . 𝒎 . 𝒎

𝒔 = 0.5916 𝟎. 𝒔

Lab Report Unit 2: FREE FALL & REACTION TIME 8

Time #2 = ∗ . 𝒎 . 𝒎

𝒔 = 0.5931 𝟎. 𝒔

Time #3 = ∗ . 𝒎 . 𝒎

𝒔 = 0.5938 𝟎. 𝒔

Time #4 = ∗ . 𝒎 . 𝒎

𝒔 = 0.5922 𝟎. 𝒔

Time #5 = ∗ . 𝒎 . 𝒎

𝒔 = 0.5947 𝟎. 𝒔

�̅� 𝒕𝒊 𝟎. 𝒔 𝟎. 𝒔 𝟎. 𝒔 𝟎. 𝒔 𝟎. 𝒔

𝒊

𝟎. 𝟎.

Finding g:

G = ∗ . 𝒎 𝟎. 𝒔

= 9.8052 . 𝟎 𝒎 𝒔

6.

Percent Error = | . 𝟎 . |

. ∗ 𝟎𝟎 𝟎. 𝟎 𝟎 𝟎. 𝟎 %

Part 2

1.

Starting from the 50cm mark.

Final Position (cm) 67.1 65 64.4 60 61

Difference #1: 67.1 – 50 = 17.1cm || Difference #1 in m = 17.1cm* 𝒎

𝟎𝟎𝒄𝒎 = 0.171m

Difference #2: 65 – 50 = 15cm || Difference #1 in m = 15cm* 𝒎

𝟎𝟎𝒄𝒎 = 0.15m

Difference #3: 64.4 – 50 = 14.4cm || Difference #1 in m = 14.4cm* 𝒎

𝟎𝟎𝒄𝒎 = 0.144m

Difference #4: 60 – 50 = 10cm || Difference #1 in m = 10cm* 𝒎

𝟎𝟎𝒄𝒎 = 0.1m

Difference #5: 61 -50 = 11cm || Difference #1 in m = 11cm* 𝒎

𝟎𝟎𝒄𝒎 = 0.11m

Lab Report Unit 2: FREE FALL & REACTION TIME 9

Average Displacement:

𝑫 ∑ 𝑫𝒊 𝒊 = 𝟎. 𝐦 𝟎. 𝐦 𝟎. 𝐦 𝟎. 𝐦 𝟎. 𝐦 = 0.135 m

Graph to Find the Relative error:

d ( ) = D 𝑫 = 0.171m-0.135m = 0.036m || |𝒅 ( ) |2 = 0.001296

d (2) = D2 𝑫 = 0.150m-0.135m = 0.015m || |𝒅 ( ) |2 = 0.000225

d (3) = D3 𝑫 = 0.144m-0.135m = 0.009m || |𝒅 ( ) |2 = 8.1*10-5

d (4) = D4 𝑫 = 0.100m-0.135m = 0.035m || |𝒅( )|2 = 0.001225

d(5)= D5 𝑫 = 0.110m-0.135m = -0.025m || |𝒅( )|2 = 0.000625

Displacement (m) Uncertainty Uncertainty ^2 0.171 0.036 0.001296

0.15 0.015 0.000225 0.144 0.009 8.1E-05

0.1 -0.035 0.001225 0.11 -0.025 0.000625

Standard Deviation:

𝑶𝒊 |𝒅𝒊| 𝑰

𝟎. 𝟎𝟎 𝟎. 𝟎𝟎𝟎 . ∗ 𝟎 𝟎. 𝟎𝟎 𝟎. 𝟎𝟎𝟎

𝟎. 𝟎 𝟎. 𝟎

Relative Error:

Lab Report Unit 2: FREE FALL & REACTION TIME 10

𝛆 𝟎. 𝟎 𝟎.

𝟎. 𝟎.

2.

Average Reaction time without air resistance:

TNoAir = ∗𝟎. 𝒎

. 𝒎 𝒔

= 0.16598 𝟎. s

3.

Average Reaction time with air resistance:

TAir = 𝟎. 𝒎

. 𝒎 𝒔

∗ . ∗ 𝟎 𝒌𝒈

𝒔 ∗𝟎. 𝒎

𝒎 ) = 0.16599 𝟎. 𝒔

4.

Internet Average Reaction = 0.215s

5.

Percentage error between my results and theorical result internet

Percentage error = |𝟎. 𝟎. |

𝟎. ∗ 𝟎𝟎 . 𝟎 . %

Conclusion:

For Part 1, I found out that the y-coordinate point when the meter stick is at the 2-meter mark is 55

mouse units, and the y-coordinate point when the meter stick is at the 0-meter mark is 500 mouse units.

Lab Report Unit 2: FREE FALL & REACTION TIME 11

The difference between both were 455 mouse units, and by rounding the value of the division between the

original value in meters by the mouse unit value, I got that a rate of 0.0045 meters per mouse unit. After

this, I got to measure 5 times the position of the ball in mouse measurements from the first picture to the

last picture. When I got the results, I calculated the difference between these 5 different positions, converted

into meters, and calculated for each trial its time, to then calculate the mean of these times, which resulted

in a mean round time of 0.593 seconds. After finding the time, I was able to find the g constant by plugging

the time in the formula 𝑔 , getting a g rounded value of 9.805 𝒎 𝒔

. Finally, I find the percentage error

of my value of g with the actual value of g (g=9.81 ) by using the formula for the percentage error

𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑒𝑟𝑟𝑜𝑟 | | ∗ 100 , getting a round value percentage error of 0.05%.

For Part 2, I recorded the 5 trials of graving the meter stick after letting it fall from the 50cm mark. After

getting all marks from where my hand stopped, I calculated the displacement by using the formula

𝑦 𝑦 𝑦 and then I converted each displacement by using the centimeter-to-meter formula

𝑥𝑐𝑚 ∗ . After this, I calculated the mean by using the formula 𝑴 𝑵

∑ 𝑴𝒊 𝑵 𝒊 , with a result of

0.135m. With this mean, I was able to calculate the graph the components to find the standard

deviation of this measurement using the formula getting a round standard

deviation of 0.026, and then I find the relative error by using the formula

𝜀 𝑶𝒊 𝑴

, getting a relative error of 𝟎. . Then, I calculated the average reaction time without

air resistance using the formula TNoAir = and with air resistance using the formula TAir = ∗

1 ∗ ). After rounding both times, I compared both times, and the difference between one

another was to small have a big impact between times, getting a rounded time for both times of

0.166 seconds. Next, I got an average reaction time from the internet of 0.215s[1], and then I

calculated the percentage error of my average reaction time with the internet average reaction time

𝑶𝒊 𝑵 |𝒅𝒊|

𝑵

𝑰

Lab Report Unit 2: FREE FALL & REACTION TIME 12

using the formula 𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑒𝑟𝑟𝑜𝑟 | | ∗ 100, getting a final percentage error of 22.8%. Finally,

I expect that the reaction time to move a leg (e.g., pressing the brake pedal in response to a visual stimulus)

is greater than the hand reaction time measured in this experiment because the distance from which

the eyes see the meter stick falling (which could be probably be calculated using a regular 12-inch

ruler) is shorter comparing to possible accident while driving (which would be no less than 1 meter

from the eye-view), therefore increasing the time our brain use to reason a physical movement.

Lab Report Unit 2: FREE FALL & REACTION TIME 13

References

[1] Human benchmark. (n.d.). Retrieved February 14, 2021, from

https://humanbenchmark.com/tests/reactiontime