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Miscellaneous Topics from mental math to quick measurement estimates.

Free Fall Distance
Estimating Free Fall Distance

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Speed of Sound
Estimating Speed of Sound

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Solar System Scale
Solar System to Scale

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Travel
Travel

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In Estimates

Quick Freefall Distance Estimates

An observer can estimate the height of a cliff by dropping a rock from the top of the cliff. The distance travelled by a free falling projectile follows the equation \(d=v_i+0.5at^2\). By applying the equation to a rock tossed from the top of a cliff with no vertical velocity component, the height of the cliff can be determined. In this scenario the distance to the bottom is \(0.5at^2\) where \(t\) is the time it takes for the rock to hit bottom and \(g\) is the gravitational acceleration \(32.174 ft/s^2\). By rounding the numbers it can be shown that the distance in feet is approximately \(16t^2\) where \(t\) is in seconds. A relatively quick way to estimate the distance in ones head is to take \(t^2\) times 10 and then add half of that number to itself. Comparisons of this approximatation method to the actual calculated distance are shown in the table below. This approximation underestimates the distance by about 7%, but manually timing the rocks free fall time is still probably a larger source of error.

\begin{equation} d=v_i+0.5at^2 \end{equation} \begin{equation} d=0.5at^2 \end{equation} \begin{equation} d=0.5*32t^2 \end{equation} \begin{equation} d=16t^2 \end{equation} \begin{equation} d=1.5*10t^2 \end{equation}

Free Fall Time (s) Distance (ft) Approximate Distance (ft)
1 16 15
2 64 60
3 145 135
4 258 240
5 403 375
6 580 540
7 789 735
8 1030 960
9 1304 1215
10 1610 1500
In Estimates

Speed of Sound Estimate

The speed of sound in air at sea level is 340.29 m/s or 0.21 miles/s. It is generally easier to remember that sound travels approximately 1 mile in 5 seconds with 5% error.

In Estimates

The Solar System to Scale

The solar system is commonly illustrated with pictures that show the planet sizes to scale, but the distances between them are not to scale or not to the same scale at least. A typical image, shown here, shows the planet sizes to scale, but the distances between them are not to scale. The video here illustrates the reason why the solar system is commonly shown this way, but this technique leads the audiance to beleive that the planets are relatively closer together than they actually are./p>

The video here shows the planet sizes, starting with the Sun, and distances between them drawn to the same scale. When the video zooms out at the end, to show the entire solar system, some planets are smaller than a single pixel when depicted on a typical monitor.