Question

1. use the distance vs. time data and a quadratic fit to determine g.
2. display the acceleration vs. time plot and note the apparent variation in acceleration. trace across the plot with the and cursor keys, and read the acceleration values individually. is the acceleration varying as much as it first appears?
3. would dropping the picket fence from higher above the photogate change any of the parameters you measured? try it.

initial velocity and time of flight may change but g should always be the same in an ideal case.

1. would throwing the picket fence downward, but letting go before it enters the photogate, change any of your measurements? how about throwing the picket fence upward? try performing these experiments.

throwing downward or upward changes initial velocity and time of flight b g is local and constant.

1. how would adding air resistance change the results? try adding a loop of clear tape to the upper end of the picket fence. drop the modified picket fence through the photogate and compare the results with your original free - fall results.

air resistance would cause a non - constant acceleration, less than g and changes distance, time, and velocity relationship.

1. investigate how the value of g varies around the world. for example, how does altitude affect the value of g? what other factors cause this acceleration to vary at different locations? how much can g vary at a location in the mountains compared to a location at sea level?

Explanation:

Step1: Determine g from distance - time data

The distance - time relationship for free - fall under constant acceleration (near the surface of the Earth, acceleration due to gravity $g$) is given by $d = v_0t+\frac{1}{2}gt^{2}$, where $d$ is the distance, $v_0$ is the initial velocity and $t$ is the time. For a free - fall starting from rest ($v_0 = 0$), $d=\frac{1}{2}gt^{2}$. By performing a quadratic fit on the distance - time data ($d$ vs $t$), the coefficient of $t^{2}$ in the quadratic equation $y = ax^{2}+bx + c$ (where $y$ is distance and $x$ is time) is $\frac{1}{2}g$. So, $g = 2a$.

Step2: Analyze acceleration - time plot

When you display the acceleration - time plot, if there is no air resistance and the motion is under the influence of gravity only, the acceleration should be a constant value equal to $g$ (in the absence of non - gravitational forces). If there are variations in the plot, it could be due to measurement errors, air resistance or other non - ideal factors. Reading the values with the cursor keys helps to quantify these variations.

Step3: Consider dropping from a higher height

Dropping the Picket Fence from a higher height above the Photogate will change the time of flight. Using the equation $d=\frac{1}{2}gt^{2}$, as $d$ increases (higher height), $t=\sqrt{\frac{2d}{g}}$ increases. The initial velocity (assuming it is dropped, $v_0 = 0$) remains the same at the start of the measurement (when it enters the Photogate's field of view), but the velocity when it reaches the Photogate will be higher ($v = gt$). However, the value of $g$ remains the same in an ideal case.

Step4: Analyze throwing the object

Throwing the Picket Fence downward or upward changes the initial velocity. If thrown downward, $v_0>0$ and if thrown upward, $v_0$ is negative (taking the downward direction as positive). The time of flight will also change. For example, for an upward - thrown object, the time of flight can be found from the equation $d=v_0t-\frac{1}{2}gt^{2}$ (using the quadratic formula $t=\frac{-v_0\pm\sqrt{v_0^{2} + 2gd}}{g}$). But $g$ is a local constant and does not change due to the initial velocity of the object.

Step5: Account for air resistance

Air resistance is a non - conservative force. When air resistance is added (by adding a loop of clear tape), the net force on the Picket Fence is $F_{net}=mg - F_{air}$, where $F_{air}$ is the air resistance force. According to Newton's second law $F = ma$, so $a=\frac{mg - F_{air}}{m}=g-\frac{F_{air}}{m}$. The acceleration is no longer constant and is less than $g$. This will change the distance - time and velocity - time relationships compared to the free - fall case.

Step6: Investigate variation of g

The value of $g$ varies with altitude ($g=\frac{GM}{(R + h)^{2}}$, where $G$ is the gravitational constant, $M$ is the mass of the Earth, $R$ is the radius of the Earth and $h$ is the altitude). As altitude $h$ increases, $g$ decreases. Other factors include the non - spherical shape of the Earth (the Earth is an oblate spheroid), and the distribution of mass within the Earth. At a mountain location compared to sea level, $g$ at the mountain is slightly less because of the increased distance from the center of the Earth.

Answer:

The steps above provide a way to solve the problems related to determining $g$ and understanding the factors affecting free - fall motion. There is no single numerical answer for all the sub - questions as they are more about the experimental procedures and understanding the concepts. For example, to find $g$ from the quadratic fit of distance - time data, you need to actually perform the fit on the given data set to get the value of the coefficient of $t^{2}$ and then calculate $g = 2a$.