Free fall is motion under gravity alone, starting from rest. Vertical throw adds initial velocity.

Free Fall (u = 0, a = g):

• v = gt, h = \frac{1}{2}$$gt^2, v = sqrt(2gh)
• Distance in nth second: $s_n$ = g$\frac{2n-1}{2}$
• Successive second ratio: 1:3:5:7:9 (Galileo's law)
• Cumulative distance ratio: 1:4:9:16:25 (perfect squares)

Vertical Throw (upward, u = $v_0$):

• Time to top: t = $\frac{v_0}{g}$
• Max height: H = $v_0$^$\frac{2}{2g}$
• Total flight time: T = 2$v_0$/g
• Symmetry: speed at any height during ascent = speed at same height during descent
• Time of ascent = time of descent (to same level)

Ball dropped vs thrown upward vs thrown downward from same height: All three hit the ground with the SAME speed (energy conservation: v = sqrt($u^2$ + 2gh)). Only the direction of impact velocity differs.