Acceleration of Gravity, by Marcus Wade

The purpose of lab 2: Acceleration of Gravity was to determine the acceleration of a freely falling object, as well as gaining more experience using the computer to capture and record data. Opening the software, there was a blank position vs. time graph that was able to be adjusted to a view screen of our choosing. A motion detector was then set up, and the logger pro software was utilized to capture and record the data for our free falling object (a rubber ball). After verifying the equipment was functioning properly, we started from a height of about 1 m, and gave the object a throw straight upwards.

Motion Detector

 After a few test runs with throwing and dropping the rubber ball, we were able to get some very nice parabolic shapes of the object's trajectory. The graph of the object's trajectory should be a parabola because it is experiencing a constant acceleration. This means that the velocity graph is also linear, and one more integration would yield a parabolic position graph. The position and velocity graphs of our first trial runs are shown.

Position Function

Velocity Function

In the position function, the x axis represents the time (seconds) and the y axis is the object's height (meters).





In the velocity function, the x axis still represents the time, but the y axis represents the speed of the falling object at that instant.

Once the position function was recorded, a quadratic curve fit was applied and and yielded the equation (-4.8x^2+14.4x-9.1). We then had to determine if the acceleration of our object matched up with the accepted 9.8 m/s^2. To do this, the coefficient (a) from our parabola was plugged into the following equation to yield our values for column 2.

G=2(a), where (a) is the leading coefficient of the parabola

G=2(-4.8)

G=-9.6 m/s^2

For column 4, our we just took the velocity graphs of each of our position graphs, and took the leading coefficient for the x variable to obtain our value for acceleration.

The following table is the data collected from each of our 5 trial runs

We calculated our percentage error from the first trial run with the following equation:

(9.6 (m/s^2) -9.8 (m/s^2))/9.8 (m/s^2)

(0.02 (m/s^2))*100= 2% error
The other experimental error values were calculated in the same ways

In this lab, we threw a ball straight up from a height of 1 m and the ball then rose about 1 m once it left our hand in the air. We were able to determine, with reasonably close accuracy, the accepted value for the acceleration of gravity. The amount of percentage error varied from greatly, from 0.2% to 8.7%. The sources of error in this experiment included various factors such as possibly not performing an exactly vertical toss, failing to adjust the y axis view screen of our velocity vs time graph to obtain a more accurate linear fit for the acceleration, and air resistance

Graphical Analysis, by Marcus Wade

The purpose of the graphical analysis lab was to basically become familiar with the logger pro software, so that we could record and interpret data for the falling object (ball). The first part of the lab was mainly to get us used to the software. After looking through a function plot graph, we chose our own function to graph:

fig 1

x*arctan(3x2)     -Graphed in fig 1







Once we became familiar with the graphical analysis software, inputting our own functions, we set up the motion detector and got ready for a few trial drops.





The run that was kept was dropped from a height of 2 meters and took approximately 0.6 seconds to fall to the ground. The equation of the object's position graph was fitted to be the parabola -4.8x^(2)+10.01x-3.211 on the domain [1,1.6]

Once the graphs had been obtained, we verified that our data matched with the equation:

d=gt^(n)

d=distance (m)

g=gravity (m/s^2)

t=time (s)

We assume the acceleration to be 9.8 m/s^2. Since the object did not start falling until t₁=1.1 s_, we subtract t₂=1.6 s from t₁=1.1 and get t=0.5 s for our equation. Also the object appeared to have fallen 1.5 m in the course of its drop.

Using dimensional analysis, we plug our values and units into the equation as follows: 

1.5 (m)=9.8 (m/s^2)*(0.5 s)^n

Dividing by 9.6, then taking logarithms of both sides of the equation to solve for n, we obtain:

log((1.5 )/(9.8))/log(0.5)=2.7 s

In order for the equation to cancel properly, n must be equal to 2. To calculate our percentage error we plug it in as follows:

((2.7 s-2 s)/2 s)*100=35% error

The purpose of this lab was quite simple, just to learn to use the logger pro software. This was achieved through creating our own graphs as well as recording position vs. time data for the falling object. Potential sources of error in this lab included fitting our graph in a bad domain, or possibly even that our object's trajectory was not exactly perpendicular to the floor.