# Non-Constant Acceleration due to Gravity

Recently, I had the first physics lab for my university physics course. This lab was fairly simple, as we were merely using a computer and a distance sensor to graph the position, velocity, and acceleration of a cart as it moved along a linear track.

One of the situations we captured data for involved starting the cart at the bottom of an inclined ramp and giving it a push upwards. As expected, it rolled up, came to a stop, and then came back down the track to its starting position. The position-vs-time graph was essentially parabolic, the velocity-vs-time graph was essentially linear, and the acceleration-vs-time graph was essentially linear. So far, so good.

At this point in the lab, the instructor pointed out that, if the data was examined closely, the acceleration of the cart was greater while the cart was traveling upwards than when the cart was traveling downwards (approximately $0.546\frac{m}{s^2}$ and $0.486\frac{m}{s^2}$, respectively), and asked us to determine why in our lab report.

Now, gravity was the only force acting upon the cart, and thus it's acceleration should be a constant, at least at the scale our experiment was conducted at, so these results are completely baffling my lab group. So far, we have proposed the following ideas, but none of them seem very plausible.

• Doppler effect on the ultrasonic distance sensor

• Friction

• Air resistance

• Human error

The first seems highly improbable, and the last three are more obfuscation and hand waving than actual theories.

Why does our experimental data show the acceleration due to gravity to change based on the direction the object is moving?

• Have you considered computing (or measuring) an estimate to as many of these as possible and seeing if any of them match the observed discrepancy? Commented Feb 1, 2014 at 0:07
• Your ultrasonic distance sensor was not measuring the acceleration due to gravity. In order to do that, the cart would need to be in free fall. Instead, your sensor was measuring the acceleration of the cart, which in your experimental setup, is not equal to the acceleration of gravity. Note that the numeric value of acceleration you measured with the sensor wasn't near $g=9.81~\text{m/s}^2$.
– BMS
Commented Feb 1, 2014 at 1:13
• @dmckee Unfortunately, I do not have access to the materials any more. I do know that the instructor asked this question to several groups in the lab, so I am assuming it was not just a fluke. Commented Feb 1, 2014 at 1:59
• @BMS Yes; bad terminology on my part. I'll edit and fix that. My intent was that the only acceleration acting upon the cart was gravity, although it is only a fraction of gravity, as it is on an inclined plane. Commented Feb 1, 2014 at 2:00
• If I may pick on a pet peeve of mine: don't ever suggest "human error" as an explanation for an observation. Human error just means someone (normally you) screwed up, and you should fix your screwup rather than simply pointing it out. (Of course I see you already share this sentiment to some extent.) Commented Feb 1, 2014 at 8:03

That the numerical value of the resulting acceleration is greater upwards than downwards is likely due to friction. Uphill friction and gravity pull in the same direction, while downhill friction and gravity pull in opposite directions.

You can use the difference in acceleration to estimate the friction. The average is the (projected) gravitational acceleration, from which you can calculate the inclination angle (assuming a constant slope).

• Thank you. This makes a lot of sense, and covers everything quite nicely. Commented Feb 1, 2014 at 2:40
• You can further test this hypothesis by attempting a fit to your data with a model that includes a constant acceleration and a friction term that's proportional to the velocity, or its square. If it works it's essentially a clincher; if you don't find a good fit it's a good guess you need to find something else. If you get a fit with large uncertainty in your parameters, though, you need to think more carefully about the whole thing. Commented Feb 1, 2014 at 19:20

The ideal case must be symmetric in time. Off by 10.7% difference/average seems significant. If the cart is not symmetric in shape, including wheel mounts, air resistance is suspect re ballistic coefficient, Cd. Does the cart have an ultrasonic reflector only on one end? Doppler shift is testable by aiming the ultrasonic sensor at the other end of the cart. Is there an air current in the room from a heater? Is there an electromagnet under the track?

When the cart went down, could air loft under its leading end? Some mgh is feeding the spinning wheels. Are they patterned in any way that air will blow differently depending on their direction of spin?

If you only ran the experiment once, you have no statistics for systematic and statistical errors. Labwork begins with "the universe hates me." Then, start shaving away footnotes. It isn't paranoia if it happens.

• "It isn't paranoia if it happens" is a great phrase. Commented Feb 1, 2014 at 1:19