## Quick Freefall Distance Estimates

Measure distance off of a cliff by dropping a rock. Distance travelled by a free falling projectile follows the equation \(d=v_i+0.5at^2\). Letting go of a rock from rest means that 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 of \(32.174 ft/s^2\). So distance in feet is approximately \(16t^2\) where t is in seconds. Quick way to do this is to do \(t^2\) times 10 and then take 1.5 times that number by adding half of it to it. Comparisons using this method to the actual calculation are shown in the table below. This approximation underestimates the distance by about 7%, but the mental measurement and rounding of \(t\) is going to approximate too so the 7% rounding probably isn't the most signifcant 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) |
Actual 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 |