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One of the most basic things students do in a physics lab is to collect data and use that to build a model. Most of these models come in the form of a mathematical function. But here is the problem. For some reasons, students dislike representing these functions graphically. They are afraid to embrace the power of the graph.

OK, let’s do a simple experiment and use a graph to find a mathematical model.

### Constant Acceleration

We are going to measure distance and time for an accelerating object and use that to find the acceleration. In the past, I would do this lab using a specialized drop timer. It was a stop watch connected to a ball dropper and a landing pad. When the ball was released, the clock would start and then it would stop when it hit the pad. You need a drop timmer for falling objects because the free fall time for an inside object is too short to accurately measure with a stop watch. Now I just use a cart rolling down an inclined track. This gives a much longer time to record the motion so that it can easily be accomplished with a stop watch.

Here you can see I have a low-friction cart on a slightly inclined track. It doesn’t really matter what angle the track is inclined, but it should stay constant. Really, this is essentially what Galileo did to investigate the acceleration of a falling object (but I guess that doesn’t really matter).

I will release the cart from rest and let it accelerate over a distance of 10 cm and record the time (I will do it 5 times to get an average and a standard deviation). After that, I will increase the starting distance and repeat it for several more distances.

If an object is moving with a constant acceleration, I can use the following kinematic equation (which I won’t derive):

In case you aren’t familiar with this equation, it basically tells you the one dimensional postion (x) for an object after some time interval (t). The x_{0} is the starting postion (at t = 0) and v_{0} is the velocity at time zero. So, for this case, I will release the cart from rest (hopefully) so that the v_{0} term will be zero. Also, I don’t really care where the cart stops or starts but just the total distance (x – x_{0}). Just to make things easier, I can consider x_{0} = 0. Now we have a simpler equation:

WARNING: Do not think of this as a fundamental equation. This is only for the special case where the object starts from rest at x = 0. OK, you have been warned. But now we have our mathematical model. As the cart accelerates through a greater distance, it will take more time. OK, let’s collect some data. Here are the rolling distances with average times and the standard deviation of times.

Don’t worry about the standard deviation stuff it it bothers you—I’m just including it for completeness. OK, we have some data, but what now? Let’s try making a graph. I am going to use plotly, but you should be able to do this on regular graph paper. There is no point using a tool if you can’t do it by hand first—so if you feel uncomfortable with graphs, use the paper.

So, here is my first plot. This has the distance on the horizontal axis and the time on the vertical (since distance is the independent variable—that’s what you would expect). Oh, don’t worry about the error-bars (the lines through the data points). I’m just including those in there for fun.

Great. We have a graph, but what do we do with it? Why should we ever make a graph? Should we just make a graph because a lab report has to have a graph? No, there is a reason to make a graph. In most cases it is to show that there is a relationship between the variables being plotted on the two axes. In this case, what do we expect? Should this be a linear function? No, our model for the acceleration does not predict that the distance should be proportional to the time. According to our kinematic equation, distance should be proportional to time squared.

Let’s make another graph. First, I am going to put distance on the vertical axis. Yes, I know that this should be on the horizontal axis since it’s the independent variable, but the graph will look better this way. Second, I want to make a graph that is linear. So let’s compare our expected model with the generic equation for a line.

As you can see, we will have to plot distance on the vertical axis to make it look like our expected linear function. For the horizontal axis, we will plot t^{2} instead of just time since the distance should be proportional to time squared.

Notice that a linear function does indeed fit this data quite nicely. But why fit a function if you don’t do something with it? In this case, the important value we need from the linear fit is the slope. If you look back at our model, you can see that we are plotting distance (x) versus time squared (t^{2}) and these two should be proportional with the constant of (1/2)a. So, the slope of our function should be (1/2)a.

Since the slope of the linear fit is 0.0541 m/s^{2} (yes, the slope has units), then the acceleration of this cart would be 0.108 m/s^{2}. Boom.

### The Common Student Method

Unfortunately, I see many students that like to approach this problem from a slightly different perspective. They will let the cart roll down the track at different starting distance and measure the time it takes. They will also do each distance 5 times—because that’s what I said (I actually say that five is the minimum). After that, they will have the same (or at least similar) distance vs. time data. But what next?

Well, let’s take one of the data points. If I let the cart roll 10 cm, it takes an average 1.378 seconds to travel. With this distance and time value, I can simply plug it into the kinematic equation and solve for the acceleration. This would give an acceleration of 0.1053 m/s^{2}. Next, I can repeat this calculation for the other distance-time values and then average all of the accelerations.

Isn’t this the same thing as making a graph? Well, no. You might get a similar value for the acceleration, but treating each point individually isn’t the same as looking at all the data at once. First, there is the model. How do you know your initial model (the kinematic equation) is legitimate if you don’t plot your data? You need to see that it sort of fits a linear function. Second, what about the y-intercept? In the linear fit above, I get a y-intercept of -0.00399 meters. This is pretty close to zero, so that’s good. But if you calculate the acceleration without the graph, you are explicitly stating that the y-intercept is zero—which it might not be.

So there are some actual reasons for making a graph. I know students often think “I have to make a graph because Dr. Allain likes graphs”—but that’s not true (well, it’s true I like graphs). You *should* make a graph because it’s probably the best way to analyze your data. You should also understand that a linear graph is nice because you can easily estimate a best fit line if you use graph paper (just by using a straight edge). Further, it is important that you find the slope and realize that this slope has some meaning. Honestly, this pops up in so many labs and students commonly struggle with this idea. I’ve gone over this before, so let me just leave you with this older post that goes over some of the details of finding the slope for a linear function.

### Another Method to Find the Acceleration

If you are a student, or just bored—feel free to stop here. You are excused. For those of you remaining, I am going to show you another way to find the acceleration from this distance-time data.

Let’s go back to our kinematic equation (assuming we start with zero velocity).

In the previous section we made this a linear function by plotting x vs t^{2}. How about not plotting a linear function? Let’s just plot x vs. t. Again, technically this should be t vs x since t is the dependent variable—but damn the rules!

Since we suspect there should be a quadratic relationship between x and t, we fit a quadratic (second order polynomial) to the data. Yes, you can’t really do this on graph paper—you essentially need a computer. I will skip the technical details of fitting a function to data since it depends on your plotting program.

The nice thing about fitting a quadratic equation is that we can throw out our assumptions of a zero starting velocity. OK, technically with our particular experiment each run has to have the same starting velocity. So really, the only way you could do this is with a zero initial velocity. However, if you use other methods to collect position-time data then there could be a non-zero starting velocity.

But how do you find the acceleration? Again, if we compare the fitting quadratic equation to the kinematic equation we see that the coefficient in from of the t^{2} term has to match up to the t^{2} term in the kinematic equation. This means that the (0.0506) in front of x^{2} in the quadratic fit must be equal to the (1/2)a term in the kinematic equation giving an acceleration of 0.1012 m/s^{2}. OK, I should point out that in many plotting programs you can change the variables in the fitting equation so that it has x and t instead of f(x) and x. I left it as x because that’s the way you often see it.

### Finding the Slope of the Incline (and Friction)

If you only care about finding the acceleration, you may be excused. If you want to stay I am going to connect the acceleration of the cart to something else—the local gravitational field.

Here is a force diagram for a cart (with no friction) rolling down an inclined plane.

Since the cart can only accelerate in the direction of the incline, there is only one force that pushes in this direction—the gravitational force. But only a component of the gravitational force accelerates the cart. The angle between this gravitational force and the y-axis (which I set as perpendicular to the plane) is the same angle (θ) that the track is inclined. This means that in the x-direction (along the plane), I have:

If I know g (the local gravitational field) and the incline of the plane (θ), I can calculate the expected value of the acceleration. The gravitational field is mostly a constant. I will use a value of g = 9.8 N/kg. For the angle, I attempted to measure this with my smartphone (with the built in level). This gave a value of 1 degree—so I suspect that this isn’t very precise. However, if I use these values in this equation I get an acceleration down the incline with a magnitude of 0.171 m/s^{2}.

That’s not good enough. How about I instead just use a better system to find the position of the cart? Here is data using Vernier’s Motion Encoder. This is basically a track with a series of lines. The cart then detects motion over these lines to give position-time data.

Again using the quadratic fit I can find the acceleration. In this case it gives a value of 0.1092 m/s^{2}. That’s pretty close to the value from my first experiment. I’m mostly happy. But what angle would this correspond to for the inclined plane? Assuming a gravitational field of 9.8 N/kg, the angle θ would have to be 0.638 degrees. So, it is entirely possible that the iPhone angle measurement just rounds up to report a tilt of 1 degree.

But what about friction? Is there a significant frictional force as the car rolls down the incline? Well, if I don’t actually know the angle of the incline it’s impossible to know if the acceleration is due to gravity alone or a combination of gravity and friction. Well, it’s impossible if you just let the cart roll down the track. However, if you let the cart go up AND down, then you can detect the frictional force. Why? Because the up acceleration should be different than the down acceleration. It will make more sense with two force diagrams.

For kinetic friction (friction between objects that move), the frictional force is in the opposite direction of motion—this is even true for a cart with wheels. So as the cart goes *up* the incline, friction is *down* the incline. This reverses as the cart goes down the incline. This means that the acceleration going up would be greater than the acceleration going down. To get a relationship between the up and down acceleration, let me start with the usual model for friction. This says that the magnitude of the frictional force is equal to the product of the normal force and some coefficient.

If I call “down” the incline the positive x-direction, then I have the following equations for the motion of the block as it goes up.

Yes, I skipped some steps—consider it homework to figure out what you missed. Also, here I am calling a_{x1} the acceleration UP the incline. Now I could do the same thing for the block sliding down the incline. The only thing that changes is the direction of the frictional force. I will call this a_{x2}.

Both accelerations have that same term due to the gravitational force. Let me subtract the down acceleration from the up acceleration.

Now that I have an expression for the coefficient of friction (μ_{k}), I can plug that back into the expression for the acceleration up the incline and then solve the angle. Yes, that seems overly complicated but it’s just another way of solving two equations. Again skipping some steps, I get the following.

So all I need to do is measure the acceleration both up and down the incline. Again, I can do that with the Vernier Encoder System. Here’s what I get.

From this you can see that the acceleration up and down the incline are indeed different (so there is friction). Up the incline I have an acceleration of 0.1435 m/s^{2} and down I get 0.10596 m/s^{2}. Putting these values into my expression for θ I get an incline of 0.529 degrees. I guess I’m happy with that. Now that I have the angle, I can solve for the coefficient of friction. I get a value of 0.0019. That’s a fairly low value for the coefficient of friction—but this is supposed to be a “low friction” track.

OK. Hopefully you have learned two things. First, graphs are important. Second, I can get a little carried away with physics—sometimes.

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