Object in free fall

After seeing this video, I decided to write a quick post to explain the simple physic involved. Everyone knows that two objects drop at the same time will reach the ground together, no matter their respective weight, but is it always true? And… why ?!

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When an object free falls on Earth, it is always subject to at least two forces, the gravitational force ($F_g$) and the drag force ($F_D$), the second is due to air resistance. From Newton’s second and famous law, we know that the sum of the forces acting on an object is equal to the mass of this object multiply by it’s acceleration.

     \begin{align*} \sum F &= ma \\ F_g - F_D &= ma \\ mg - \frac{1}{2} C_D \rho U^2 A & = ma \end{align*}

When drag force isn’t negligible, free falling object will eventually reach a maximum velocity called the terminal velocity. This happen because the drag force increases with the squared of the velocity of the falling object. It can increase until it is equal to the gravitational force.

At that point, the net force on the body is zero (no more acceleration!) and the terminal velocity is:

     \begin{align*} \sum F &= 0\\ mg - \frac{1}{2} C_D \rho U^2 A & = 0\\ U &= \sqrt{\frac{2mg}{C_D \rho A}}. \end{align*}

As we can see from last equation, if we compare two object of the same shape, the heaviest will fall at a higher speed and reach ground first. However from lower height and using aerodynamic object, the drag force can be considered negligible compared to the gravitational force. This leads to the following equation:

     \begin{align*} \sum F &= ma \\ F_g &= ma \\ mg & = ma \\ g &= a. \end{align*}

In this case the acceleration of the free failing object ($a$) is equal to the Earth gravity ($g$), independently of its mass.

Another way of demonstrating this phenomenon, is to completely remove the air in the room. No air $\to$ no drag! This is exactly what Brian Cox did when he visited NASA’s Space Power Facility in Ohio, shown in this great video from the BBC.

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