Vertical Throw & Free Fall Solver Tutorial

What Is Vertical Throw / Free Fall?

Vertical throw and free fall describe one-dimensional motion (straight up or down) under constant gravitational acceleration. In the ideal model only gravity acts; air resistance, wind and spin are ignored.

Free fall: v₀ = 0 — object released from rest, gravity only.
Downward throw: v₀ < 0 — initial velocity directed downward.
Upward throw: v₀ > 0 — object rises, slows, reaches peak (v = 0), then falls.

Sign Convention

This calculator uses upward as positive. Therefore:

Enter g as a positive number, e.g. 9.81 m/s²
The acceleration in the equations is −g (downward is negative)
Upward throw: v₀ > 0; downward throw: v₀ < 0
Impact velocity is negative (object moving downward) — its magnitude |v| is shown separately

Core Equations

1 – Position equation

y(t) = h₀ + v₀·t − ½·g·t²

h₀: initial height (m) · v₀: initial velocity (m/s) · g: gravity (m/s²) · t: time (s)

2 – Velocity equation

v(t) = v₀ − g·t

Velocity decreases by g each second. In free fall, downward speed increases by g per second.

3 – Velocity–position relation (time-independent)

v² = v₀² − 2·g·(y − h₀)

Find velocity at any height without knowing time.

How Is Impact Time Found?

Setting ground level at y = 0 and solving y(t) = 0 gives a quadratic equation in t:

0 = h₀ + v₀·t − ½·g·t²
→ (−½g)·t² + v₀·t + h₀ = 0

Root: t = (−v₀ ± √(v₀² + 2·g·h₀)) / (−g)
The physically valid root is t > 0 (the smallest positive root).

If the discriminant D < 0, there is no real root — the object never reaches y = 0 under these conditions.

Maximum Height (Upward Throw Only, v₀ > 0)

The object decelerates; when velocity reaches zero it is at the peak:

v(t_max) = 0 → t_max = v₀ / g
h_max = h₀ + v₀² / (2·g)

Example: h₀ = 0, v₀ = 20 m/s, g = 9.81 m/s² → t_max ≈ 2.04 s, h_max ≈ 20.39 m

Example: Free Fall (h₀ = 45 m, v₀ = 0)

y(t) = 45 − ½ · 9.81 · t²
0 = 45 − 4.905·t² → t = √(45/4.905) ≈ 3.03 s
v_impact = −9.81 · 3.03 ≈ −29.7 m/s (|v| ≈ 29.7 m/s)

What Does the Time Table Show?

The table acts like a film strip of the motion: at each time step Δt, height y(t) and velocity v(t) are listed. This makes it easy to see:

y(t) changes parabolically — curving downward (or first upward then downward).
v(t) changes linearly — confirming constant acceleration.
At the peak, velocity passes through zero and becomes negative (downward).

How Accurate Is This Model in the Real World?

No air resistance: Real objects slow down due to drag; light/large objects (e.g. feathers) are most affected.
Constant g: Valid near Earth's surface; g varies very slightly at high altitudes.
One dimension: Horizontal motion not included; use a projectile calculator for oblique throws.

Despite these simplifications, this model is the standard first step in engineering and physics education.

Note: This calculator is intended for educational and preliminary estimation purposes. It assumes no air resistance and constant gravity.

Vertical Throw & Free Fall Solver – Tutorial