Free Fall Calculator
Enter any two of the four free-fall variables — fall distance, initial velocity, time, or impact velocity — and the calculator finds the rest. The standard setup is "dropped from rest" with downward as the positive direction. The underlying math is the same four kinematic equations applied to motion with \(a = g\).
A free-falling object accelerates downward at a constant \(g \approx 9.8\) m/s² (on Earth, ignoring air resistance). It's a special case of constant-acceleration motion: the four kinematic equations apply with \(a = g\). Knowing any two of fall distance, initial velocity, time, or impact velocity is enough to find the rest.
The Formulas (Taking Downward as Positive)
Free fall uses the standard kinematic equations with \(a = g\):
| Equation | Use when… |
|---|---|
| \(v = v_0 + gt\) | You know two of \(v\), \(v_0\), \(t\) |
| \(h = v_0 t + \tfrac{1}{2} g t^2\) | You want \(h\) given \(v_0\) and \(t\) |
| \(v^2 = v_0^2 + 2gh\) | You want \(v\) given \(v_0\) and \(h\) (no time involved) |
| \(h = \tfrac{1}{2}(v_0 + v) t\) | You want \(h\) from average velocity and time |
For an object dropped from rest (the most common setup), \(v_0 = 0\) and the formulas simplify:
- \(v = gt\)
- \(h = \tfrac{1}{2} g t^2\)
- \(v^2 = 2gh\), so \(v = \sqrt{2gh}\) and \(t = \sqrt{2h/g}\)
Worked Example: dropped from a cliff
A rock is dropped from a \(45\) m cliff. How long does it fall, and how fast is it going when it hits?
Starting from rest, \(v_0 = 0\). For time:
\[h = \tfrac{1}{2} g t^2 \quad \Rightarrow \quad t = \sqrt{\dfrac{2h}{g}} = \sqrt{\dfrac{90}{9.8}} \approx 3.03 \text{ s}\]
For impact velocity:
\[v = \sqrt{2 g h} = \sqrt{2 \cdot 9.8 \cdot 45} \approx 29.7 \text{ m/s}\]
Tips for Using the Calculator
- Downward is positive in this calculator. A ball thrown straight up would have a negative \(v_0\) — but for that scenario, the more general Kinematics Calculator is easier.
- For a "thrown downward" problem (initial speed pointing down), enter a positive \(v_0\).
- Air resistance is ignored. For dense objects falling short distances this is a good approximation. For very long falls or low-density objects (feathers, balloons), the real impact velocity will be much lower than the calculator predicts because air resistance becomes significant.
- On other worlds, change the gravity field. The Moon is \(1.625\) m/s²; Mars is \(3.71\) m/s²; Jupiter is \(24.8\) m/s².
Frequently Asked Questions
What's the difference between "free fall" and other falling motion?
In physics, free fall specifically means motion where gravity is the only force. An object falling through air with appreciable drag isn't strictly in free fall, even though the language outside physics often uses the term loosely. For introductory problems where air resistance is ignored, free fall just means "falling under gravity alone."
What's terminal velocity?
Terminal velocity is the speed at which air resistance balances gravity, so the object stops accelerating and falls at a constant speed. It depends on the object's shape, mass, and air density. A skydiver in a belly-down position reaches terminal velocity at roughly \(55\) m/s; a small dense object like a steel ball would need to fall much farther before air resistance matched its weight. This calculator ignores air resistance entirely, so it doesn't model terminal velocity.
Why does a heavier object fall at the same rate as a lighter one?
In free fall (no air resistance), the acceleration is the same for every object regardless of mass. Mathematically, Newton's second law gives \(F = ma\); for gravity, \(F = mg\), so \(a = g\) — the mass cancels out. Air resistance is what causes lighter or larger-surface objects to fall more slowly in real life.
What happens to the velocity if I throw the object downward first?
Set \(v_0\) to a positive value (this calculator uses downward as positive). The object will reach the ground sooner and faster than if dropped from rest. The same kinematic equations still apply.
Can I use this for an object thrown upward?
You can, but it's easier to switch to the Kinematics Calculator for that case — it doesn't have a built-in "downward is positive" assumption. If you do use this one, enter \(v_0\) as a negative number and take any "fall distance" \(h\) above the launch point as negative.