Forces of Nature Lab

Uniform motion in one dimention, whether or not it is under constant velocity, is difficult to observe carefully. You must be able to mark the position of the falling object without actually affecting the motion of the object, so somewhat more sophisticated lab equipment is required than you usually find around the house.

In a standard AP class or freshman college physics lab, you would perform experiments and record the position with respect to time of a little metal cart on an air track. Unfortunately, the link to the high school class report on linear track experiments no longer works, so I'll have to hunt up another-watch this space.

We are going to use a less precise method of determining acceleration during free fall. Rather than try to determine the position of our falling body at successive seconds during a single fall, we are going to vary the drop distance and time the body's descent multiple times. In doing this, we have to assume that the forces acting on the ball do not change, and that we can keep from introducing factors (differences in air friction, changes in mass, changes in horizontal velocity) which might cause the motion to vary from one drop to the next.

You are at liberty to use any other means to measure the vertical distance; this is only a suggestion.

- String or nylon, 10m. Knot the string every meter.
- Some way to mark the string (magic marker, stickies, tape).
- Weight for string (to keep it taut while measuring vertical distance).
- Location which varies drop distance (at least 10 feet). The stairwell of a two story building is perfect if you can get permission to perform the drop. Higher is better, but be careful about hitting innocent non-physicists or your partner.
*PARTNER*. You will need someone to help you time the fall.- Stopwatch, perferably with 0.1 accuracy.
- Weight. A rubber ball is good, since it generally does little damage on impact, although you have to be spry to catch it.
- The usually paper and pencil to record your drops.

*Data Collection*

- Find the location and secure permission to perform you experiments from the appropriate authorities.
- Starting at the top or bottom of your range, determine how many measurments you want to take and how far apart they should be. For example, if you have a 10M range, you can take measurements every 2 meters to get 5 distance points. You may have to vary the distances if you can't easily make them even; that is okay.
- Describe your weight. If you use a ball, record its diameter, mass, anything you notice about it that may affect its fall rate.
- At each distance from the ground
- record the distance to the ground (be sure to specify units!)
- drop your weight and time the fall at least three times. Be sure to drop the weight using the same starting position for each fall (bottom of weight, usually).
- record each drop time to the nearest tenth of a second if possible.

*Data Reduction*

Arrange your data as you perform the reduction in some neat order, so that it is easy to see and understand what you have done. You may want to consider setting up a spreadsheet and letting it do the calculations for you. Here is a suggestion, but you can improve on it.

Distance | Drop1 | Drop2 | Drop3 | Average | Velocity(mps) | Acceleration |

4m | 1.0 | 1.0 | .9 | 1.0 | ||

8m | 1.3 | 1.3 | 1.3 | 1.3 | ||

12m | 1.6 | 1.6 | 1.5 | 1.6 | ||

16m | 1.8 | 1.9 | 1.8 | 1.8 | ||

20m | 2.1 | 2.2 | 2.2 | 2.167 |

You will also need to determine the accuracy of your measurements. Physicists use a statistical method called determination of the standard deviation. According to this theory, 68.3% of all repeated measurments should fall within the standard deviation (plus or minus) from the average.

$$\begin{array}{l}\sigma \text{}=\sqrt{\frac{1}{n-1}\sum _{i}^{n}{\text{}[{x}_{i}\text{}-\text{}\overline{x}]}^{2}}\text{}\\ \sigma \text{}=\text{}\mathrm{standard}\text{}\mathrm{deviation}\\ n\text{}=\text{}\mathrm{number}\text{}\mathrm{of}\text{}\mathrm{data}\text{}\mathrm{points}\\ i\text{}=\text{}\mathrm{the}\text{}\mathrm{ith}\text{}\mathrm{individual}\text{}\mathrm{data}\text{}\mathrm{point}\end{array}$$In the table above, the standard deviation at 20m is the square root of (1/3) * ((2.17 -2.2)^{2} + (2.2-2.17)^{2} + (2.2-2.17)^{2}) = .0577 ~ .06. So about 70% of my measurements should be within .06 sec of the average 2.167--which they are.

- For each distance, determine an average time based on your three measurments.
- Determine the standard deviation for the measurement. If you have a scienctific calculator, you may follow the instructions for determining standard deviation on it.
- Plot your average drop distance data distance vs. time. What is the shape of the line?
- Determine the velocity for each measurement. Remember that distance over total time will give you an average velocity. How did you determine the instantaneous velocity at some point?
- Plot the velocity as a function of time. What shape is the line?
- Determine the acceleration. Is it changing?

Your report should include:

- A description of your equipment and procedures which is sufficiently detailed that I could repeat your experiment myself to check your results.
- Your raw data and calculated data, with explanations of your asusmptions and calculations.
- Your conclusions about acceleration: is it constant?

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