Example

Question:

Free-fall: Discuss the motion of an object under free fall. Neglect air resistance.

Solution:

An object released near the surface of the Earth is accelerated downward under gravity.
The magnitude of acceleration due to gravity is represented by \( g \). If air resistance is neglected, the object undergoes free fall.
For small heights, \( g \) can be taken as constant, \( g = 9.8~\mathrm{m/s^2} \).

Motion is in \( y \)-direction, but since gravity is downward, \( a = -g = -9.8~\mathrm{m/s^2} \).
The object is released from rest at \( y = 0 \), so \( v_0 = 0 \), and the motion is described by:

• \( v = 0 - g t = -9.8 t~\mathrm{m/s} \)
• \( y = 0 - \frac{1}{2} g t^2 = -4.9 t^2~\mathrm{m} \)
• \( v^2 = 0 - 2 g y = -19.6 y \; \mathrm{(m^2/s^2)} \)

These equations give the velocity and the distance travelled as a function of time, and also the variation of velocity with distance. The variation of acceleration, velocity, and distance with time are plotted as shown below:
Summary:

• Acceleration: constant, \( a = -g \), does not change with time (a horizontal line).
• Velocity: decreases linearly (negative slope) with time.
• Distance: increases quadratically with time in the negative \( y \)-direction.