Determining g: Evaluating Methods of Measuring Acceleration

Please ignore the bouncing title. Like seriously. It’s not even proper HTML.

What’s the Big Idea‽

How does gravity affect the motion of objects? The force of gravity is consistently affecting everything on our planet, but its effect becomes even more apparent in falling objects. Gravity causes all objects to undergo a constant acceleration of 9.8 m/s²—but how can we calculate this value (notated as g) using only our knowledge of kinematics? 

We completed two different labs to find out and then determined which method was better.

Procedure

The first lab was the Picket Fence Free Fall lab, which required a photogate, a LabQuest, and, of course, a picket fence (which is a thick, transparent strip of plastic with black rectangles evenly space apart). There were also some additionally materials such as the stand and the cloth to make for a soft landing. The resulting setup looked like this:

Once everything was connected, we dropped the picket fence as indicated by the diagram. One of our trials looked like this:

Notice the quadratic curve in the distance-vs.-time graph, a hallmark of constant acceleration. The velocity-vs.-time graph is linear, because the velocity increased at a constant rate. We repeated the procedure several times to get the following composite graph:

Each trial resulted in similar graphs that matched the first.

Now, on to the second lab: the Ball Toss lab! The materials were simple: a ball, a motion detector, and once again, a Vernier LabQuest.

The procedure, like the materials, was also simple: just hold the ball in front of the motion detector, throw it up, and catch it before it hits the motion detector. However, I found this to be rather clumsy because it was difficult to throw the ball straight up and have it return to the starting position. Our most successful trial yielded the following graph:

Like the first lab, we can see the quadratic curve in the first graph and the linear one in the second—giving us insight into how gravity affects an object. The velocity graphs of both experiments happen to be particular insightful—notice the value for m. The first experiment yields a value of 9.666 and the second experiment yields a value of -8.953. Though the signs of these values aren’t important, their magnitudes are. Roughly speaking, they match the Earth’s value of acceleration due to gravity—9.8 m/s² downwards.

Takeaways

If you haven’t realized it yet, we performed these experiments on Earth. Meaning our predicted values for the acceleration due to gravity were fairly accurate, however more so in the Picket Fence Free Fall lab. These predicted values are the criteria by which we are judging the two labs.

There are a couple reasons why I think the first experiment resulted a value closer to the one we were looking for. This concerns accuracy and precision. Accuracy is generally associated with systemic error, whereas precision is generally associated with random error. We were able to eliminate much more (but certainly not all) systemic and random error in the Picket Fence Free Fall lab, meaning our value of 9.666 was was both more accurate and more precise than the one we got in the Ball Toss lab. We were able to repeat the first experiment multiple times without changing the result very much, which ultimately allowed us to average our data (minimizing random error and increasing precision). Furthermore, dropping the picket fence was much less awkward and clumsy than throwing the ball up, which explains the difference in accuracy (the first experiment was affected by less systemic error than the second lab).

Ultimately, I felt that the tools (i.e. the photogate and the motion detector) were both fairly reliable and not a major source of error in either experiment.

And for gravity? Well, we were able to visualize its affects on an object in free fall graphically, as well as roughly determine its value based on our knowledge of kinematics.

Video Analysis of Motion

What’s the Big Idea‽

Motion. But this time, not on an incline, but rather, graphed. How does constant velocity and constant acceleration look on a graph? Allons-y !

Procedure

The procedure for this lab was quite simple. Take one video of something exhibiting a constant velocity, another of something exhibiting constant acceleration, and analyze them using Vernier Graphical Analysis.

We represented constant velocity with the tumble buggies, since their velocity changed very little, and constant acceleration with a water droplet, since its velocity increased at a constant rate due to the force of gravity. Though the videos have been lost to the mysteries of existence, the screenshots of our analyses remain intact.

Both graphs show a change in position over time, i.e. velocity. However, since the velocity in the first graph is constant, the graph is linear. The velocity in the second graph is not constant (but the acceleration is), so the graph is quadratic. Notice how acceleration is literally position divided by time squared.

Takeaways

This lab reinforces the graphical visualisation of kinematics. In order to understand the underlying principles in physics, it is important to understand and interpret graphs for different kinds of motion.

Motion on an Incline

What’s the Big Idea‽

Motion. On an incline, this time. Not only are we dealing with just one dimension of motion, but two dimensions of motion! In order to understand how linear kinematics work, it is necessary to see what velocity and acceleration look like and how motion in different directions affect each other. (Spoiler: not at all!)

Procedure

For this laboratory, we needed a track, a low-friction cart, a set of physics books, a laptop, and a Vernier LabQuest and a corresponding motion detector. We set up the track with the motion detector at the end, then used the books to elevate the track (that’s really all they’re good for now, since we switched to the iPads). The LabQuest and laptop recorded the data. We placed the cart at the lower end of the track and pushed with moderate force, much like this image suggests:

The push resulted in the cart being pushed up the ramp, until it stopped and rolled back down towards the starting position. The resulting data is as so:

Both graphs are annotated, but I’ll explain them a bit. For reference, each “section” is delineated by yellow lines.The first graph shows position vs. time (i.e. velocity) and the second graph shows velocity vs. time (i.e. acceleration). 

1. In the first section, notice how, on both graphs, the line is different from the rest of the graph. This is the when we were pushing the cart in both the x-direction (when accounting for the ramp angle); it was receiving a net positive acceleration. 
2. The second section shows the cart going up, whereas the third section shows it going down. On the first graph, these to sections make up a quadratic curve; this is gravity doing its work in conjunction with a change in x position (though gravity isn’t acting strictly in the y-direction, in this case). The second graph is also insightful—it’s linear! This is clearly because of gravity’s constant downward force on the cart.
3. The last section shows another little wonky line, like the first section. This is the cart coming into contact with the motion detector.

In addition to the normal procedure of the lab, we played around with the cart, utilizing its spring function. The result was interesting:

For this lab, it doesn’t have that much significance and would most likely be more confusing without additional physics knowledge. But I’m keeping it here regardless of that fact, just because I like it.

Takeaways

There are two main takeaways in the lab. Firstly, the visualization of velocity and acceleration. In different situations, these characteristics of motion look different in real life than they do on a graph. It is important to comprehend graphs in order to get a deeper understanding of physics. Secondly, though I don’t think this lab explicitly dealt with x- and the y-direction motion, I was able to notice the constant acceleration due to gravity, both in the quadratic curve of the first graph and the linear fit of the second graph. Again, this is a vital concept in the comprehension of physics and the world around us.