The vertical coordinate \( y(t) \) of an object in free fall can be described using the following kinematic equation, assuming that the object is falling under the influence of gravity and air resistance is negligible:
\[ y(t) = y_0 + v_0 t + \frac{1}{2} g t^2 \]
where:
- \( y(t) \) is the vertical position of the object at time \( t \),
- \( y_0 \) is the initial vertical position (starting height),
- \( v_0 \) is the initial vertical velocity (positive if directed upwards and negative if directed downwards),
- \( g \) is the acceleration due to gravity (approximately \( 9.81 , \text{m/s}^2 \) downward),
- \( t \) is the time in seconds since the object began its fall.
If the object is dropped (i.e., released from rest), then \( v_0 = 0 \), and the equation simplifies to:
\[ y(t) = y_0 + \frac{1}{2} g t^2 \]
Make sure to take the sign convention into account, as \( g \) is usually treated as a negative value when considering downward motion from the initial height.
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