Free Fall With Air Resistance Calculator

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Model a straight-down drop or downward throw with air resistance (downward is positive) and get time, impact speed, terminal velocity, and a no-drag comparison.

Choose the main thing you want: time to fall a given height, height fallen after a given time, or speed after a given time. The calculator will compute the rest too.
Quadratic drag is the usual model for objects falling through air at everyday speeds. Linear drag is more appropriate for very slow motion in very thick fluids (like glycerin) or very small particles.
Vertical distance from start to impact. Use meters. Example: 100 m is about a 30-story building. Must be > 0 for time-to-fall mode.
Mass of the object. Needed for drag motion because heavier objects are less slowed by drag.
Area facing the airflow. For a human belly-to-earth, a rough range is around 0.5 to 0.8 m^2. For a ball, you can compute area with A = pi * r^2.
Advanced options
Environment and forces
Shape factor for quadratic drag. Typical values: smooth sphere about 0.47, flat plate broadside about 1.2 to 1.3, human belly-to-earth about 1.0.
Density of air. Sea level standard air is about 1.225 kg/m^3. Higher altitude usually means lower density and less drag.
Acceleration due to gravity. Standard gravity is 9.80665 m/s^2.
Buoyancy is the upward force from displaced air: F_b = rho_air * volume * g. In air it is usually small unless the object is very low density or very large.
Only used if buoyancy is included. Volume of the object. For a sphere: V = (4/3) * pi * r^3.
Start and model-specific input
Starting speed downward. Use 0 for a simple drop. Use a positive number if the object is thrown downward. Negative means an upward throw.
Only used for Linear drag model. Drag force is F_d = k * v. If you do not know k, use Quadratic drag instead.
Extra targets and display
Used in modes that need time, like height after a given time or speed after a given time.
Optional. If provided, the calculator will also estimate the time and distance needed to reach this speed (if reachable under the model).
Choose how results are shown: Auto, Plain, or Scientific. Auto uses scientific notation only for very large or very small values.
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How to use our Free Fall With Air Resistance Calculator

  1. Pick What do you want to calculate? (time to fall a height, height after a time, or speed after a time).
  2. Choose a Drag model: use Quadratic for most real falling in air; use Linear only when you have a value for Linear drag constant k (N*s/m).
  3. Enter Mass (kg) and Cross-sectional area (m^2).
  4. If your mode is about impact, enter Drop height (m). If your mode is time-based, open Advanced options and enter Time (s).
  5. Open Advanced options and (for Quadratic drag) set Drag coefficient Cd (unitless) and Air density (kg/m^3) if you need something other than the defaults.
  6. Set Initial downward velocity (m/s): use 0 for a simple drop, positive for a downward throw, and negative for an upward throw.
  7. Optional: set Target downward speed (m/s) if you want the calculator to report whether that speed is reachable and (if so) about when and how far it takes.
  8. Click Calculate.
  9. Sanity-check: for a drop or downward throw, Time to impact with drag should usually be longer than Time to impact (no air resistance), and Speed at impact (downward) should usually be less than Impact speed (no air resistance). Also compare Speed at impact (downward) to Terminal velocity to see if the object got close to its speed limit.

Definitions

Downward positive: The sign rule used here. Downward velocity is positive, and upward velocity is negative.

Drag (air resistance): A force from moving through air that pushes opposite the motion and reduces acceleration.

Quadratic drag: A drag model where the drag force grows with speed squared (v^2). This is the usual choice for objects moving through air at everyday speeds.

Linear drag: A drag model where the drag force grows directly with speed (v). It is mainly used for very slow motion or very thick (viscous) fluids; it requires Linear drag constant k (N*s/m) [1].

Terminal velocity: The steady falling speed where weight (minus buoyancy if included) balances drag, so the net acceleration is about zero [2].

Velocity vs speed: Velocity includes direction (can be positive or negative). Speed is the size of the velocity (never negative) [3].

Common mistakes and quick fixes

Mistake: Using Quadratic drag but entering 0 (or a negative number) for Drag coefficient Cd (unitless) , Air density (kg/m^3) , or Cross-sectional area (m^2) .
Fix: For Quadratic drag, all three must be > 0. If you cannot estimate them, switch to a no-drag comparison or use Linear only if you know Linear drag constant k (N*s/m) .

Mistake: Choosing Linear drag but leaving Linear drag constant k (N*s/m) at 0.
Fix: Enter k > 0, or switch back to Quadratic drag. Linear drag results are not meaningful without k.

Mistake: Entering an upward throw as a positive Initial downward velocity (m/s) .
Fix: This calculator uses downward as positive, so an upward throw must be a negative number.

Mistake: Turning on Include buoyancy? but guessing a volume that makes buoyancy almost as big as weight.
Fix: If buoyancy is on, use a realistic Object volume (m^3) . If the calculator warns that effective gravity is near zero or negative, buoyancy is too large for the given mass and volume (or the object would float).

Mistake: Setting Target downward speed (m/s) at or above Terminal velocity and expecting a finite time and distance to reach it.
Fix: Pick a target below terminal velocity, or change inputs that increase terminal velocity (larger mass, smaller area, smaller Cd, lower air density). At terminal velocity, acceleration goes to about zero, so the exact terminal speed is not reached in finite time in this simple model [2].

Mistake: Confusing Speed at impact (downward) with Velocity at impact (downward positive) when Initial downward velocity (m/s) is negative.
Fix: Use Speed for magnitude (always nonnegative). Use Velocity for direction (signed) under the downward-positive rule [3].

Methodology

Sign convention: Downward is positive. The calculator tracks y (downward distance fallen, in m) and v (downward velocity, in m/s). Impact is when y reaches Drop height (m).

Forces: Weight acts downward. Drag acts opposite the motion (so during a downward fall it acts upward). If enabled, buoyancy also acts upward.

m * dv/dt = m*g - F_drag - F_buoy

dy/dt = v

Buoyancy (optional, constant):

F_buoy = rho_air * volume * g

Drag models:

Quadratic drag (typical in air):

F_drag = (1/2) * rho_air * Cd * area * v^2

Linear drag (special-case):

F_drag = linear_k * v

Terminal velocity: Terminal velocity is the steady speed where dv/dt = 0, meaning drag balances the effective downward pull. Define effective gravity as g_eff = g - (F_buoy/m). Terminal velocity is a key idea for falls with drag [2].

Quadratic drag terminal speed:

v_t = sqrt( (2*m*g_eff) / (rho_air*Cd*area) )

Linear drag terminal speed:

v_t = (m*g_eff) / linear_k

Numerical solve (how the calculator gets time and distance with drag): With drag, there is not one simple kinematics formula that covers all inputs. The calculator steps forward in small time steps, updating v from dv/dt and y from dy/dt, until it reaches the requested target (impact height, target time, or target speed). If v0 is negative (thrown upward), it continues through the upward part, past the stop point (v = 0), and into the downward fall.

No-drag comparison: For the same gravity and initial velocity, the calculator also shows ideal results with no air resistance:

v(t) = v0 + g*t

y(t) = v0*t + (1/2)*g*t^2

v^2 = v0^2 + 2*g*y

Worked mini-example (Quadratic drag): Use Drop height = 100 m, Mass = 80 kg, Area = 0.7 m^2, Cd = 1.0, Air density = 1.225 kg/m^3, g = 9.80665 m/s^2, v0 = 0, buoyancy off. First compute terminal velocity (so you know the speed limit):

v_t = sqrt((2*80*9.80665)/(1.225*1.0*0.7)) = about 42.7 m/s

Then the solver steps forward until y = 100 m to get time to impact and impact speed, and reports percent of terminal reached as 100 * impact_speed / v_t.

Limitations & Key Assumptions / Boundary Conditions

This is a 1D vertical model only. It does not include sideways motion, changing body position, lift, spin effects, or wind gusts.

The Quadratic model assumes Air density (kg/m^3), Drag coefficient Cd (unitless), and Cross-sectional area (m^2) stay constant during the fall. In real life, density changes with altitude and weather, and Cd and area can change with orientation and speed.

The Linear model is a special-case approximation and depends on the correct value of Linear drag constant k (N*s/m). Using Linear drag with an unknown or guessed k can give very wrong answers.

Buoyancy (if enabled) is treated as a constant upward force using your Air density (kg/m^3) and Object volume (m^3). If buoyancy is close to weight, the calculator may show that the object does not fall downward (effective gravity near zero or negative).

Drag motion results use numerical stepping (small time steps) with safety limits. For extreme inputs (very large height, tiny mass, huge area, very low effective gravity, or long upward-then-downward motion), the solver may stop early and return N/A with a warning.

The Energy lost to drag (approx) output is model-based energy bookkeeping, not a measured heat or sound value, and it ignores effects like deformation at impact.

Sources

  1. Terminal Velocity - Falling with a Linear Drag Force - Tugraz
  2. Terminal velocity | Definition, Examples, & Facts | Britannica - Encyclopedia Britannica
  3. Velocity | Speed, Acceleration, Motion | Britannica - Encyclopedia Britannica
  4. [1305.1283] An analytic solution to the equations governing the motion of a point mass with quadratic resistance and generalizations - Arxiv