Speed and acceleration in free fall

Relationship between velocity and height of fall

It is sometimes useful to work out the velocity of an object in terms of the distance travelled h, without involving time. We use the equation:

v² = v₀² + 2g h

where h is the height of the fall (vertical distance travelled) and v₀ is the initial velocity.

Example: an object dropped with no initial velocity (v₀ = 0) falls a distance of h = 5 m. v² = 2 × 9.8 × 5 = 98, so v = √98 ≈ 9.9 m/s.

Graphical representation

The graph of v(t) is a straight line with a slope of g (since v(t) = g*t + v0). The graph of y(t) is a parabola because y depends on t². These two graphs allow us to interpret information visually: the slope of the v(t) graph gives the acceleration.

Graph	Shape	What can be read
v(t)	straight line	slope = g
y(t)	parabola	curvature related to g

Common pitfall

Be careful with units: g must be expressed in m/s² and distances in metres in the formulae. Convert time to seconds and height from centimetres to metres before carrying out any calculations; otherwise, the result will be incorrect by a factor of 10, 100 or more.