Time Equation for Free Fall- Physics Formulas Explained
What Is the Time Equation for Free Fall?
The time equation for free fall tells you how long an object takes to hit the ground when dropped from a certain height. It's one of the first equations you learn in physics, and it's surprisingly useful in real life.
The basic formula is:
t = √(2h/g)
Where:
- t = time in seconds
- h = height in meters (or feet, depending on your units)
- g = gravitational acceleration ≈ 9.8 m/s² on Earth
That's it. No complicated calculus. No hidden tricks. Just square root, division, and multiplication.
The Derivation (Keep It Simple)
You get this equation from two basic kinematic equations:
1. Distance: h = ½gt²
2. Solve for t: t = √(2h/g)
Air resistance is ignored here. That means this formula only works perfectly in a vacuum. In real life, a feather and a bowling ball don't fall at the same speed—but they would on the Moon.
Variables You Need to Understand
Gravitational Acceleration (g)
This value changes depending on where you are:
- Earth: 9.8 m/s²
- Moon: 1.6 m/s²
- Mars: 3.7 m/s²
- Jupiter: 24.8 m/s²
Use the right value for your situation. Plugging in Earth's gravity for a Moon calculation gives you garbage results.
Height (h)
Measure from the release point to the ground. If you're dropping something from a 50-meter building, h = 50 meters. The object doesn't start at ground level.
How to Solve Free Fall Time Problems
Step 1: Identify your known values (height and local gravity)
Step 2: Plug them into t = √(2h/g)
Step 3: Calculate the numerator (2h)
Step 4: Divide by g
Step 5: Take the square root
Example Calculation
Drop a ball from 20 meters. How long until it hits the ground?
t = √(2 × 20 / 9.8)
t = √(40 / 9.8)
t = √(4.08)
t ≈ 2.02 seconds
That's the answer. No extra steps needed.
Common Mistakes to Avoid
- Using the wrong units — mix meters and feet and you'll get nonsense
- Forgetting to square root — the formula requires it
- Including initial velocity — this formula assumes the object starts from rest
- Ignoring air resistance — this formula is theoretical, not practical for parachutes or paper airplanes
Time vs. Height Comparison
| Height (m) | Time to Fall (seconds) |
|---|---|
| 5 | 1.01 |
| 10 | 1.43 |
| 20 | 2.02 |
| 50 | 3.19 |
| 100 | 4.52 |
| 500 | 10.10 |
Notice: time doesn't scale linearly with height. Double the height, and time increases by roughly 1.4x (because of the square root).
What If the Object Is Thrown Downward?
The simple formula assumes zero initial velocity. If you throw the object downward with speed v₀, use this instead:
t = (-v₀ + √(v₀² + 2gh)) / g
Or solve the quadratic equation h = v₀t + ½gt² for t. The negative root gives you the physically meaningful answer.
What If the Object Is Thrown Upward?
That's not free fall—that's projectile motion. The object goes up, stops, then falls back down. Total time in the air depends on the initial upward velocity, not just the height.
For an object thrown upward from height h with velocity v₀:
Total time = (v₀/g) + √(2h/g + (v₀/g)²)
This gets messy fast. Most introductory physics problems keep it simple with pure vertical drops.
Free Fall on Different Planets
The same equation applies. Just change the g value:
| Celestial Body | g (m/s²) | Time from 100m |
|---|---|---|
| Earth | 9.8 | 4.52s |
| Moon | 1.6 | 11.2s |
| Mars | 3.7 | 7.36s |
| Jupiter | 24.8 | 2.84s |
| Sun | 274 | 1.71s |
Drop something on the Sun and it hits the surface in under 2 seconds. You won't be there to see it, but the math is solid.
Real-World Applications
- Construction safety — calculating how long a dropped tool takes to reach workers below
- Forensics — determining if a fall was survivable based on height
- Sports science — analyzing jump times and hang time in basketball
- Engineering — designing parachute deployment sequences
Quick Reference: The Formula in Different Forms
| What You Know | Formula |
|---|---|
| Height, want time | t = √(2h/g) |
| Time, want height | h = ½gt² |
| Height, want final velocity | v = √(2gh) |
| Time, want final velocity | v = gt |
These four equations cover 95% of basic free fall problems. Memorize them or keep this page bookmarked.
The Bottom Line
The time equation for free fall is t = √(2h/g). Plug in your height, divide by gravity, take the square root. That's the whole process.
Don't overthink it. Don't look for hidden complexity. The formula works because gravity is consistent—every object at a given height falls in the same time, assuming you ignore air resistance.
For most practical problems, that's close enough.