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.
4. 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.
5. 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.
6. 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

Explanation:

Step1: Determine g using quadratic fit

The distance - time relationship for free - fall is $d = v_0t+\frac{1}{2}gt^{2}$, where $v_0$ is the initial velocity. If the object is dropped ($v_0 = 0$), $d=\frac{1}{2}gt^{2}$. Performing a quadratic fit on the distance - time data ($d$ vs $t$) gives a function of the form $y = ax^{2}+bx + c$. When $v_0 = 0$, the coefficient $a$ of the $x^{2}$ term is related to $g$ by $a=\frac{1}{2}g$, so $g = 2a$.

Step2: Analyze acceleration - time plot

When looking at the acceleration - time plot, apparent variations may be due to measurement errors, air resistance (if not negligible), or inaccuracies in the data - collection device. Reading the acceleration values with the cursor keys can help determine if the variation is significant. In an ideal free - fall situation, the acceleration should be constant ($g$ near the Earth's surface), but in reality, factors like air resistance can cause fluctuations.

Step3: Consider height above photogate

Dropping the picket fence from higher above the photogate may change the initial velocity measured by the photogate if there is air resistance or if the object gains speed over the extra distance. In an ideal no - air - resistance case, as long as the photogate measures the initial conditions accurately, the value of $g$ determined should be the same, but the time of flight and the velocity when passing through the photogate will be different.

Step4: Analyze throwing the picket fence

Throwing the picket fence downward before it enters the photogate gives it an initial velocity. This will change the measured velocity at the photogate and the time of flight through the photogate. Throwing it upward will also change the initial conditions. In both cases, the measured values related to velocity and time will be affected, but the value of $g$ itself (the acceleration due to gravity) remains the same in the local environment, provided air resistance and other non - ideal factors are accounted for.

Step5: Analyze effect of air resistance

Adding air resistance (by adding a loop of clear tape) will cause the object to have a non - constant acceleration. The object will reach a terminal velocity if dropped from a high enough height. The measured acceleration will be less than $g$ and will vary with time as the object's speed changes. Comparing with the original free - fall results (no air resistance), the distance - time and velocity - time relationships will be different.

Step6: Investigate variation of g

The value of $g$ varies around the world. Altitude affects $g$ because $g=\frac{GM}{(R + h)^{2}}$, 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. Other factors include the local density of the Earth's crust (more dense regions can cause a slightly higher $g$) and the Earth's rotation (which has a small effect on the measured value of $g$ at different latitudes). In the mountains, $g$ is slightly less than at sea - level because of the increased distance from the Earth's center.

Answer:

1. Perform quadratic fit on $d$ vs $t$ data, $g = 2a$ where $a$ is the coefficient of $t^{2}$ term in the fit.
2. Read acceleration values from the plot. Variations may be due to non - ideal factors like air resistance and measurement errors.
3. Initial velocity and time of flight may change, but $g$ should be the same in an ideal case.
4. Throwing downward or upward changes initial velocity, affecting measured velocity and time of flight, but $g$ is local and constant (ignoring non - idealities).
5. Air resistance causes non - constant acceleration, less than $g$, and changes distance - time and velocity - time relationships.
6. $g$ varies with altitude ($g=\frac{GM}{(R + h)^{2}}$), local density, and Earth's rotation. It is less in the mountains compared to sea - level.