Calculation of the Free Fall

If we know the local gravity respectively the gravitational acceleration, the acceleration due to gravity or the location factor of a location, we can easily derive from this information some answers to questions about the free fall, such as the speed of fall, the duration of the fall or the distance covered.

How and with which formulas we can calculate these parameters, I would like to show you in this info, using the following examples:

• Fall Time for a certain Height
• Final Speed after falling from a certain Height
• Speed after a certain Period of Falling
• Distance travelled after a certain Fall Duration

The local gravity of our home planet Earth is 9.832 m/s² at the poles, while it is slightly lower at 9.787 m/s² at the equator. In our examples - as it is usual in physics - we are using a gravitational acceleration of 9.81 m/s², the average of the Earth's surface. In my article about the local gravity of different places and planets, you will find an overview of other local gravities if you want to calculate the free fall for other conditions, such as for the moon or other planets in our solar system. In addition, we are ignoring the air resistance in our calculations, as this is of little importance during the first few meters of the fall of compact objects and only plays an increasingly important role at higher falling speeds.

Fall Time for a certain Height

First, let’s look at the formula for the duration of a fall from a given height:

\[t=\sqrt{\frac{2\cdot h}{g}}\]

With this, we can, for example, calculate how long a fall from a height of 10 meters would take on the Earth by using "10 m" for h and our local gravity for the Earth's surface for g:

\[t=\sqrt{\frac{2\cdot 10 m}{9.81\frac{m}{s^2}}}\thickapprox 1.4278 s \]

The result are rounded 1.4278 seconds. So, if we jump from the ten-meter tower into the swimming pool, it will take us just about one and a half seconds to complete this distance.

Final Speed after falling from a certain Height

Of course, we are also interested in the maximum speed we reach shortly before our impact. This can be easily calculated using the following formula:

\[v=\sqrt{2\cdot g\cdot h}\]

Again, we just need to insert our local gravity as well as the height for g respectively h. For our example of jumping from the ten-meter tower, this results in the following calculation:

\[v=\sqrt{2 * 9.81\frac{m}{s^2} * 10 m} \thickapprox 14.01 \frac{m}{s} \thickapprox 50.43 \frac{km}{h} \]

The formula gives us the result in meters per second (m/s). If we prefer the more common unit kilometers per hour (km/h) instead, we can easily convert the result of the calculation by multiplying the outcome by a factor of 3.6. This factor is obtained by the conversion of 60 * 60 = 3600 seconds for one hour divided by 1000 meters for one kilometer.

So, when we hit the water after our ten-meter jump, we have reached a final speed of approximately 14 meters per second respectively around 50 km/h, just before we enter the water.

Speed after a certain Period of Falling

Of course, we can not only calculate the final speed but also the speed we have reached after a certain amount of time. The formula for this is:

\[v=g*t\]

In addition to our local gravity g, we can now insert the time for t into the formula. If we wanted to stick with our example, we could, for example, look at how fast we would be traveling after falling for one second. However, this calculation would be trivial, since the speed in m/s for a one-second fall would correspond to the location factor (9.81 m/s or around 35 km/h), which is why we want to use a different value this time and calculate the speed reached after a free fall of 5 seconds:

\[v=9.81\frac{m}{s^2}*5s = 49.05 \frac{m}{s} = 176.58 \frac{km}{h} \]

The result shows us that after five seconds of free fall, without taking air resistance into account, we have reached a speed of 49.05 m/s respectively 176.58 km/h. As explained in the last section, we can easily calculate the km/h value by multiplying the m/s value by a factor of 3.6.

Distance travelled after a certain Fall Duration

Finally, we are interested in the distance we have covered after a certain duration of fall. We can calculate this using the following formula:

\[s=\frac{1}{2}g*t^2\]

As an example, we would like to use the 5 seconds from the example calculation from the last section again. So, we insert this time for t next to the gravitational acceleration g into the formula:

\[s=\frac{1}{2}*9.81\frac{m}{s^2}*5s^2 = 122.625 m \]

As a result we get 122.625 meters. This means that after 5 seconds of free fall we have already covered a distance of around 122 meters and we are traveling at around 176 km/h. This height approximately corresponds to the world's highest freefall tower, "Zumanjaro: Drop of Doom" at Six Flags Great Adventure Park in Jackson Township, New Jersey, USA, which has a height of 126 meters.