Finding gravity through experimental data

[BayernBlues]

Homework Statement

Experimentally measured time of fall
vs. distance for a freely falling object
Distance, x (m) Time, t (s)
0.1 0.148
0.2 0.196
0.3 0.244
0.4 0.290
0.5 0.315
0.6 0.352
0.7 0.385
0.8 0.403
0.9 0.429
1.0 0.451
Let the mathematical model be: x = 0.5gt^2 and assume that x = 0 at t = 0 :
a. In the light of what you learned in problem 1.1 (parts a and b) and the form of the equation
x = 0.5gt^2 , discuss (without calculations) two different methods that can be used to find g, the
acceleration due to gravity, from a straight line graph representation. In each method, indicate
explicitly the vertical variable, horizontal variable, slope, and intercept. Draw sketches.
b. Find a numerical estimate for g based on the above table, using the EXCEL program?

Homework Equations

x = 0.5gt^2

The Attempt at a Solution

I know that the value of g is 9.81 m/s^2 but this is asking how to show it through experimental data and graph. I think maybe taking the ln of both sides to change this into y=mx+b form might help but am unsure. Also, I thing g would be the constant in the equation if it was changed to y = mx + b form therefore it'd be the slope value.

 

[Gib Z]

It gives you the values of x and t, you have the equation that relates them. Solve for g.

 

[ogb p]

I would approach this problem by first understanding the physical concept of gravity and how it relates to the mathematical model provided. From the given equation, we can see that the distance traveled by the object (x) is directly proportional to the square of the time (t) it takes to fall. This means that as the time increases, the distance traveled increases at a faster rate.

Method 1:
One method to find the acceleration due to gravity (g) from the given data is by plotting a graph of distance (x) versus time squared (t^2). This will result in a straight line with a slope of 0.5g. The vertical variable in this method would be the distance (x) and the horizontal variable would be the time squared (t^2). The slope of the line represents 0.5g, and the intercept at x=0 represents 0. This method allows us to directly read off the value of g from the slope of the line.

Method 2:
Another method to find g would be to plot a graph of distance (x) versus time (t). This will result in a parabolic curve, as seen in the given data. However, by taking the inverse of both sides of the equation x = 0.5gt^2, we get t = √(2x/g), which is in the form of a straight line with a slope of √(2/g). In this method, the vertical variable would be the time (t) and the horizontal variable would be the distance (x). The slope of the line represents √(2/g), and the intercept at x=0 represents 0. By finding the slope of the line and solving for g, we can obtain the value of g.

To find a numerical estimate for g using the given data, we can use the Excel program to plot a graph of distance (x) versus time squared (t^2) and find the slope of the line. This will give us a value of 9.81 m/s^2, which is the accepted value of g. This shows that our experimental data is consistent with the known value of g and supports the validity of our mathematical model.