Free Fall & Gravity Calculator

Free Fall Explained: Formulas, Assumptions, and Quick Checks

Free fall describes vertical motion under a constant gravitational acceleration g with no air resistance. In this idealized model, the only force acting on the object is gravity, so its vertical position follows a simple quadratic curve in time. This calculator focuses on three classroom staples—time to hit the ground, impact velocity, and kinetic energy—and lets you explore different worlds using presets for Earth, Moon, Mars, and Jupiter.

Core model

We measure height from the ground upward and take upward as the positive direction. With initial height h₀ (meters), initial vertical velocity v₀ (m/s), and gravity magnitude g (m/s²) acting downward, the height at time t seconds is y(t) = h₀ + v₀ t − (g/2) t². The time to impact is the non-negative solution of y(t)=0:

t_hit = ( v₀ + √( v₀² + 2 g h₀ ) ) / g.

The impact velocity (signed, upward positive) is v_impact = v₀ − g t_hit, and the impact speed is |v_impact|. Kinetic energy on impact is KE = ½ m |v_impact|²; if mass is not provided, the calculator reports KE per kilogram (J/kg), which equals ½ |v|².

What changes between planets?

Only g changes. On the Moon (≈1.62 m/s²), falls are slower and impact speeds are lower for the same height. On Mars (≈3.71 m/s²) they are in between Earth (≈9.80665 m/s²) and the Moon. Jupiter’s strong gravity (≈24.79 m/s²) produces much shorter fall times and higher impact speeds. Use the presets to compare how g reshapes the height–time curve and your results.

Common pitfalls (and how to avoid them)

• Sign convention: Enter v₀ positive for an upward toss, negative for a downward throw. The tool handles the signs internally.
• Initial height = 0: With h₀=0 and v₀=0, the time is zero because you are already at the ground. Use a positive height to see a fall.
• Units: All inputs are SI (m, s, kg). Keep everything consistent to avoid conversion errors.
• No air resistance: Real objects experience drag, which typically increases fall time and reduces impact speed compared to this ideal model.

Worked example

Drop (no initial push) from h₀=10 m on Earth (g=9.80665). Then t_hit = √(2h/g) ≈ √(20/9.80665) ≈ 1.43 s. Impact speed is |v| ≈ g·t ≈ 9.80665 × 1.43 ≈ 14.0 m/s. If the mass is 2 kg, the kinetic energy at impact is ½·2·14.0² ≈ 196 J. Changing only g to the Moon stretches the time to ≈3.5 s and lowers the speed to ≈5.7 m/s.

Why the quadratic?

Acceleration due to gravity is (approximately) constant near a planet’s surface. Integrating constant acceleration gives a linear velocity change in time and a quadratic position curve. That’s why the height–time graph is a downward-opening parabola whose curvature increases with larger g.

Tip: Use a small positive v₀ to model tossing an object upward before it falls. The calculator automatically includes the up-and-down portion in the total time to impact.