In this lab, we will investigate how to measure position, velocity, and acceleration. A good source for constant acceleration is gravity. But acceleration due to gravity is quite large. It is about 9.8 m/s (or meters-per-second, per second). That means that our velocity increases about 10 m/s per second. So, for example, after free-falling for 3 seconds, an object’s velocity would be about 30 m/s, or roughly 67 mph (108 kph). In just 3 seconds! Imagine if your car could accelerate that fast.
In order to make lab measurements on position, velocity, and acceleration, we need to measure the time for each action. At high speeds, events are over very quickly. It is very difficult to click a stopwatch on and off to measure a time of about one second. So we have two needs: (1) A way to easily time short intervals, and (2) a way to reduce gravity’s acceleration. Read on.
The first need is accurate timing. The tool we will use for timing is your (or your parent’s) smartphone. Start an audio recording, verbally call out points in time, then use an audio editor to measure the times from the recording. You could also do this with video, but it’s harder. I’ll explain how to do this a little later. On an Android phone, I used a free app called “AuphonicEdit.” I’m sure that the iPhone has several to choose from. If you find one that works well, please let me know and I’ll mention it here.
The second need we have is, to reduce the acceleration of gravity. We can do this by using an inclined plane. While I’d like to take credit for this idea, it’s actually a technique that Galileo used in the 15th century. Since gravity always acts straight down, and our plane is inclined at some other angle, we can break the acceleration of gravity into two components — parallel to the plane and perpendicular to the plane. We’ll go into some detail how to calculate these components later (using trig), but right now we’ll just “reason our way through it.”
Let’s look at the two extreme cases. If set up our plane so that it was vertical, then the angle between it and vertical would be 0 Conversely, if our plane was horizontal, it makes an angle of 90 from vertical.
In the vertical case, the plane has no effect on the acceleration of gravity, so the coefficient (fraction) of gravity is 1.0. In the second case, we get no effect from gravity, so the coefficient of is 0 — no acceleration. So write this as a function
Now we need to figure out which trig function gives us these values. In other words, what trig function gives us 1 at and 0 at ? The answer is cosine.
So now we can write the equation that describes the fraction of gravity given to us by the inclined plane: where is the angle of the plane measured from vertical.
Let’s calculate a few common angles
You can use an inclinometer app on your smartphone to measure the angle of the plane. Just beware that the angle it gives you is measured from the horizontal, so you will need to take its complement to use our formula. For example, if your phone told you that the plane was at an angle of then you have and So we now have a “device” that can give us any fraction of that we desire. Handy!
Next, set up a plane. I used a flattened piece of cardboard with one end taped to the floor, leaning on a box. Measure the angle, and remember to take the complement of this angle (see above). Find a ball — I used a tennis ball. Measure from the bottom of the incline to a wall or other stopping point. Call this distance
Doing the Experiment
When the ball is rolling down the incline, it has positive acceleration (in the amount ). When it reaches the floor, it has zero acceleration (constant velocity). Note the time at these three points: Time of release (), when the ball reaches the floor (), and when it has traveled some distance, ().
Start your audio recorder app and narrate these points. Say, “one, two, three”. Then bring the recording up in an audio editor and measure the time between the spoken words, indicated by the sound waveform.
Repeat this experiment several times, trying: different incline angles, different distances for and different balls.
We have measured three times, so we can calculate two intervals. The duration of acceleration is given by The duration of zero acceleration is given by
Above we figured out that the acceleration of gravity on an inclined plane is If we multiply that times the duration of the acceleration, we get final velocity — the velocity at the bottom of the plane, given by
During the time of zero acceleration, this velocity should stay constant across the floor. We can also measure the velocity as distance divided by duration, given by
Do this experiment several times, and save the numbers. In a later lab we will talk about statistics and how to reduce experimental error.
Another source of error is friction. We will deal with this also in a later lab.
- Do you get the same time measurements each run? Why or why not?
- Do you get the same velocity from both calculations — that is, does ? Why or why not?
- Does the angle of the incline make a difference in their agreement?
- Do the two techniques for measuring velocity agree better with a shorter or longer distance, ?
- What changes do you find by using different balls?
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