AcademyOne-Dimensional Motion
Academy
Free-Fall Motion
Level 1 - Physics topic page in One-Dimensional Motion.
Principle
Free fall is one-dimensional motion driven only by gravity. Near Earth, the model uses constant downward acceleration and ignores air resistance.
Notation
\(y\)
vertical position
\(\mathrm{m}\)
\(y_0\)
initial vertical position
\(\mathrm{m}\)
\(v_y\)
vertical velocity
\(\mathrm{m\,s^{-1}}\)
\(v_{0y}\)
initial vertical velocity
\(\mathrm{m\,s^{-1}}\)
\(a_y\)
vertical acceleration
\(\mathrm{m\,s^{-2}}\)
\(g\)
gravitational field strength near Earth
\(\mathrm{m\,s^{-2}}\)
\(t\)
elapsed time
\(\mathrm{s}\)
Method
Free fall is the constant-acceleration model with gravity supplying the same downward acceleration throughout the motion.
Choose upward positive
\[a_y=-g\]
Gravity points downward, opposite the positive direction.
Integrate acceleration
\[v_y=v_{0y}-gt\]
Integrate velocity
\[y=y_0+v_{0y}t-\frac{1}{2}gt^2\]
Eliminate time
\[v_y^2=v_{0y}^2-2g(y-y_0)\]
Use this when the problem gives positions but not the time interval.
At the peak
\[v_y=0\]
This is a velocity condition, not an acceleration condition.
The plotted trajectory uses upward as positive. The curve bends downward throughout because the acceleration is negative at every point, including the peak.
At the peak, only the velocity is zero. The acceleration remains downward.
Rules
These are the compact results from the method above.
Acceleration
\[a_y=-g\]
Velocity
\[v_y=v_{0y}-gt\]
Position
\[y=y_0+v_{0y}t-\frac{1}{2}gt^2\]
No-time form
\[v_y^2=v_{0y}^2-2g(y-y_0)\]
Near Earth
\[g\approx9.81\,\text{m s}^{-2}\]
Examples
Question
An object is dropped from height \(h\). How long does it take to reach the ground?
Answer
Take upward as positive with ground at
\[y=0\]
Then \[y_0=h\]
\[v_{0y}=0\]
and \[a_y=-g\]
\[0=h-\frac{1}{2}gt^2\]
\[t=\sqrt{\frac{2h}{g}}\]
Checks
- Gravity points downward.
- Mass does not appear in the ideal free-fall equations.
- Air resistance is not included.
- At peak height, velocity is zero but acceleration is still downward.