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Free Fall: How Bodies Fall Downwards Due to Gravity

Important Formula

Formula: Free Fall

$$g ~=~ \frac{F_{\text g}}{\class{brown}{m}}$$ $$g ~=~ \frac{F_{\text g}}{\class{brown}{m}}$$ $$F_{\text g} = \class{brown}{m} \, g$$ $$\class{brown}{m} ~=~ \frac{F_{\text g}}{g}$$

Rearrange formula

What do the formula symbols mean?

Gravitational acceleration

$$ g $$ Unit $$ \frac{\mathrm{m}}{\mathrm{s}^2} $$

On earth it has approximately the value $ g = 9.8 \, \frac{\mathrm{m}}{\mathrm{s}^2} $. On Jupiter this acceleration of a body towards Jupiter would be larger, because Jupiter has a much larger mass.

Gravitational force

$$ F_\text{g} $$ Unit $$ \mathrm{N} $$

It is the force that pulls all bodies to the ground.

Mass

$$ \class{brown}{m} $$ Unit $$ \mathrm{kg} $$

Mass of the body on which the gravitational force is exerted.

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Free fall is a motion in which a body with mass $ \class{brown}{m} $ is only subject to the influence of the gravitational force $ F_{\text g} $. No other forces, such as those due to air resistance or buoyancy, are acting. In free fall, a body experiences only the gravitational acceleration $ g $, which accelerates the body towards the ground.

$$ \begin{align} g ~=~ \frac{F_{\text g}}{\class{brown}{m}} \end{align} $$

Gravitational Force (Weight) Acts on a Mass

$$ \begin{align} y(t) ~=~ \frac{1}{2} \, g \, t^2 \end{align} $$

Distance Traveled in Free Fall from Zero Starting Height

$$ \begin{align} \class{blue}{v} ~=~ \sqrt{2g \, (h_0 ~-~ h)} \end{align} $$

Free fall with air resistance

Without air resistance, the body would accelerate continuously until it hits the ground. The air resistance results in a constant final speed during free fall.

When the body falls, its speed increases. However, as the speed increases, the force $F_{\text s}$, which is caused by air resistance and slows the body down, also increases:

$$ \begin{align} F_{\text s} ~=~ \frac{1}{2} \, C \, \rho \, A \, v^2 \end{align} $$

As soon as the force of air resistance and the opposing falling force $F_{\text g} = m \, g $ add up to zero, a balance of forces is established:

$$ \begin{align} F_{\text g} ~=~ F_{\text S} \end{align} $$

Anders ausgedrückt: Es wirken keine Kräfte auf den Körper mehr, und er fällt mit einer konstanten Geschwindigkeit $v$ zu Boden, die aus der Fallkraft $m\,g$ sowie 4 und 4 berechnet werden kann:

$$ \begin{align} v ~=~ \sqrt{ \frac{2m \, g}{C \, \rho \, A} } \end{align} $$