Free Fall Calculator: How to Solve Time, Velocity, and Height Step by Step
A free fall calculator finds the time, velocity, or height of an object falling only under the pull of gravity, using the free fall formula h = ½gt² and v = gt, where g is the acceleration due to gravity (about 9.8 m/s² near Earth's surface). Unlike general projectile motion, true free fall assumes no horizontal velocity and no air resistance, so the motion is purely vertical and depends on just a handful of variables: initial height, time, initial velocity, and final velocity. This guide covers where the free fall equations come from, how to rearrange them to solve for any missing variable, two fully worked numeric examples, the mistakes that most often break a free fall calculation, and practice problems with checked answers.
Contents
- 01What Is a Free Fall Calculator and What Formulas Does It Use?
- 02How Do You Derive the Free Fall Formulas from Newton's Second Law?
- 03How Do You Rearrange the Free Fall Formula to Solve for Time, Velocity, or Height?
- 04Worked Example 1: How Long Does It Take a Ball Dropped From a Building to Hit the Ground?
- 05Worked Example 2: What Initial Height Produced a Measured Impact Velocity?
- 06What Common Mistakes Break a Free Fall Calculator Result?
- 07Practice Problems: Can You Solve These Free Fall Problems?
- 08How Is Free Fall Different From Projectile Motion, and How Can Solvify Help You Check Your Work?
What Is a Free Fall Calculator and What Formulas Does It Use?
A free fall calculator applies the constant-acceleration equations of motion to an object that starts at rest (or with a known initial velocity) and falls under gravity alone, with air resistance ignored. Three formulas cover every free fall scenario: the height formula h = ½gt² (or h = v₀t + ½gt² with an initial velocity), the velocity formula v = gt (or v = v₀ + gt), and the velocity-height relationship v² = 2gh (or v² = v₀² + 2gh). Because free fall only involves vertical motion along a straight line, these are the standard kinematics equations with the general acceleration term a replaced by g, and no horizontal component to track. Near Earth's surface, g is approximately 9.8 m/s² (9.81 m/s² for more precision, or 32.2 ft/s² in imperial units), and it is treated as constant for any fall of a few hundred meters or less — the small variation in g with altitude and latitude doesn't matter until distances become planetary in scale. A free fall calculator takes whichever variables are known — height, time, initial velocity, or final velocity — and solves for whatever is missing using the appropriate rearrangement of these three formulas.
Every free fall calculation reduces to three formulas — h = ½gt², v = gt, and v² = 2gh — with g ≈ 9.8 m/s² standing in for the constant acceleration of gravity.
How Do You Derive the Free Fall Formulas from Newton's Second Law?
The free fall formulas aren't arbitrary — they follow directly from Newton's second law and the definition of constant acceleration, which is why the same free fall equation works for a dropped coin, a skydiver before the parachute opens, or a stone thrown off a cliff.
1. Step 1 — Start from Newton's second law
For an object falling freely, the only force acting on it is gravity, so F = mg. Newton's second law states F = ma, so mg = ma, and the mass cancels to give a = g — every falling object accelerates at the same rate regardless of its mass, as long as air resistance is negligible.
2. Step 2 — Integrate acceleration to get velocity
Since a = g is constant, integrating with respect to time gives v = v₀ + gt, where v₀ is the initial velocity (v₀ = 0 for an object that is simply dropped).
3. Step 3 — Integrate velocity to get position
Integrating v = v₀ + gt with respect to time gives h = v₀t + ½gt², the total distance fallen after time t.
4. Step 4 — Eliminate time to relate velocity and height directly
Solving h = v₀t + ½gt² for t and substituting it back into v = v₀ + gt produces v² = v₀² + 2gh, which is useful whenever time isn't known or asked for.
a = g falls straight out of F = ma once gravity is the only force acting — mass cancels, which is why a feather and a bowling ball accelerate identically in a vacuum.
How Do You Rearrange the Free Fall Formula to Solve for Time, Velocity, or Height?
Every free fall calculator problem asks for one of four unknowns — time, velocity, height, or initial velocity — and each has a direct formula once you know which two variables are already given. A free fall calculator simply picks the rearrangement that matches the two known values and skips the algebra of getting there, but understanding each rearrangement is what lets you spot a typo in the input or a suspicious output before trusting the number.
1. Solving for time given height (dropped from rest)
t = √(2h/g). This only applies when v₀ = 0; if the object was thrown rather than dropped, use the quadratic h = v₀t + ½gt² instead.
2. Solving for time given velocity
t = v/g when v₀ = 0, or more generally t = (v − v₀)/g.
3. Solving for velocity given height
v = √(2gh) when v₀ = 0, or v = √(v₀² + 2gh) in general.
4. Solving for height given time or velocity
h = ½gt² when v₀ = 0, or h = v²/(2g) when v₀ = 0 and only the final velocity is known.
Once you know which two of height, time, and velocity are given, the free fall formula only has one variable left to isolate.
Worked Example 1: How Long Does It Take a Ball Dropped From a Building to Hit the Ground?
A ball is dropped (v₀ = 0) from the roof of a 45.0 m building. How long does it take to hit the ground, and how fast is it moving at impact? Use g = 9.8 m/s².
1. Step 1 — List the known values
h = 45.0 m, v₀ = 0, g = 9.8 m/s². Two unknowns are requested: time t and final velocity v.
2. Step 2 — Solve for time using t = √(2h/g)
t = √(2 × 45.0 / 9.8) = √(90.0 / 9.8) = √9.184 = 3.03 s.
3. Step 3 — Solve for velocity using v = gt
v = 9.8 × 3.03 = 29.7 m/s.
4. Step 4 — Check the answer with v = √(2gh)
v = √(2 × 9.8 × 45.0) = √882 = 29.7 m/s, which matches the Step 3 result and confirms both the time and velocity are correct.
3.03 s and 29.7 m/s — two formulas, h = ½gt² and v = √(2gh), cross-check each other whenever both time and velocity are asked for in the same problem.
Worked Example 2: What Initial Height Produced a Measured Impact Velocity?
A dropped object hits the ground at 24.5 m/s. From what height was it released, and how long did the fall take? Use g = 9.8 m/s².
1. Step 1 — List the known values
v = 24.5 m/s, v₀ = 0, g = 9.8 m/s². Two unknowns are requested: height h and time t.
2. Step 2 — Solve for height using h = v²/(2g)
h = 24.5² / (2 × 9.8) = 600.25 / 19.6 = 30.6 m.
3. Step 3 — Solve for time using t = v/g
t = 24.5 / 9.8 = 2.50 s.
4. Step 4 — Check the answer with h = ½gt²
h = 0.5 × 9.8 × 2.50² = 0.5 × 9.8 × 6.25 = 30.6 m, which matches the Step 2 result and confirms the height and time are consistent.
30.6 m and 2.50 s — working backward from a measured impact velocity uses the exact same three formulas, just solved for different unknowns.
What Common Mistakes Break a Free Fall Calculator Result?
Most incorrect free fall calculator answers come from a handful of predictable setup errors rather than a misunderstanding of the physics itself.
1. Assuming v₀ = 0 when the object was thrown, not dropped
The simple formulas h = ½gt² and v = gt only apply to an object released from rest. If the object was thrown downward or upward, the initial velocity term v₀ must be included, and the height equation becomes the quadratic h = v₀t + ½gt².
2. Mixing metric and imperial units for g
g is 9.8 m/s² in metric units but 32.2 ft/s² in imperial units. Plugging a height measured in feet into a formula using g = 9.8 m/s² (or vice versa) produces an answer that's wrong by a large, confusing factor.
3. Losing track of sign convention
If upward is chosen as positive, then g and downward velocities must be entered as negative values, and an object thrown upward has a positive v₀ that decreases to zero before becoming negative on the way down. Switching sign conventions mid-problem is one of the most common sources of error.
4. Applying the free fall formula where air resistance actually matters
These equations assume negligible air resistance, which is a good approximation for dense, compact objects falling short distances (a dropped stone or ball) but breaks down for light or fast-falling objects like a sheet of paper, a feather, or a skydiver approaching terminal velocity.
5. Rounding g or intermediate results too early
Using g = 10 m/s² as a shortcut, or rounding an intermediate square root to only 1 or 2 significant figures, can shift the final answer noticeably. Keep at least 3 significant figures through the calculation and round only at the end.
6. Confusing height fallen with height above ground
If an object is dropped from a platform 10.0 m above a 5.0 m wall, the free fall equations use the distance actually fallen (10.0 m), not the platform's height above some other reference point. Always confirm what h represents before substituting it into a free fall calculator.
Matching units, tracking whether v₀ is really zero, and picking a consistent sign convention are what separate a correct free fall calculation from one that's off by a sign or a factor of 3.28.
Practice Problems: Can You Solve These Free Fall Problems?
Work through each problem by hand using g = 9.8 m/s² before checking the answer, then use a free fall calculator only to confirm the result once you've committed to a setup.
1. Problem 1 — Find time and impact velocity
A stone is dropped from a 20.0 m cliff. Find the time to reach the bottom and its impact velocity. Answer: t = √(2 × 20.0 / 9.8) = √4.08 = 2.02 s; v = gt = 9.8 × 2.02 = 19.8 m/s.
2. Problem 2 — Find height from impact velocity
An object hits the ground at 39.2 m/s after being dropped. Find the height it fell from. Answer: h = v²/(2g) = 39.2² / 19.6 = 1536.64 / 19.6 = 78.4 m.
3. Problem 3 — Find time with a nonzero initial velocity
A ball is thrown downward with an initial velocity of 5.00 m/s from a 60.0 m tower. Find the time to hit the ground. Answer: solve 60.0 = 5.00t + 4.90t², which rearranges to 4.90t² + 5.00t − 60.0 = 0. Using the quadratic formula, t = [−5.00 + √(25.0 + 4 × 4.90 × 60.0)] / (2 × 4.90) = [−5.00 + √1201] / 9.80 = [−5.00 + 34.66] / 9.80 = 3.03 s.
4. Problem 4 — Find distance and velocity from time
A rock falls from rest for 4.00 s. Find the distance fallen and its final velocity. Answer: h = ½ × 9.8 × 4.00² = 0.5 × 9.8 × 16.0 = 78.4 m; v = gt = 9.8 × 4.00 = 39.2 m/s.
If Problem 3 came out near 3.03 s, the initial downward velocity was correctly folded into the quadratic instead of being dropped — that's the step that trips up most first attempts at a thrown-object free fall problem.
How Is Free Fall Different From Projectile Motion, and How Can Solvify Help You Check Your Work?
Free fall is really just the vertical special case of projectile motion, where the horizontal velocity is zero. Full projectile motion — a ball thrown at an angle, a launched cannonball — splits into two independent parts: a horizontal component that moves at constant velocity, and a vertical component that follows exactly the same free fall equations covered here, h = ½gt² and v = v₀ + gt, with g pulling it back down. Once the vertical half of a projectile motion problem is set up correctly, it's solved with the same free fall calculator steps used for a straight vertical drop. Solvify's step-by-step physics solver works like a free fall calculator that shows the full substitution and arithmetic behind every h, t, and v result, so you can line up your handwritten work against the calculator's steps and find exactly where a mistake happened instead of just seeing a different final number. A practical workflow: solve the problem by hand first, decide explicitly whether v₀ is zero, and pick a sign convention before substituting numbers. Then compare your answer against the calculator, and if the numbers don't match, walk through its steps to find the exact line where your setup diverged — usually a missed initial velocity term, a unit mismatch, or a sign error.
Free fall is just projectile motion with the horizontal velocity set to zero — master the vertical half here, and you already know most of what projectile motion requires.
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