Projectile Motion Calculator with Air Resistance

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Real balls do not follow a perfect parabola: air drag shortens the range and lowers the top. This calculator steps through the flight with a quadratic drag force and compares it with the no-drag answer.

Standards:

Our alignment; not reviewed by the standards bodies.

Formulas verified in build:

Calculator

Results

  • Range with drag93.9 m
  • Highest point with drag20.2 m
  • Flight time with drag4.05 s
  • Range without air153.3 m
  • Highest point without air26.8 m

Results update as you type.

The equation

m v⃗˙=mg⃗−12ρCdA ∣v⃗∣ v⃗,A=πd24m\,\dot{\vec v}=m\vec g-\tfrac12\rho C_d A\,|\vec v|\,\vec v,\quad A=\tfrac{\pi d^2}{4}
R0=v02sin⁡2θg  (no drag)R_0=\frac{v_0^2\sin 2\theta}{g}\ \ (\text{no drag})

Worked example

Given
  • Launch speed v₀ (m/s, up to 200): 40 m/s
  • Launch angle θ (°): 35 °
  • Mass (kg): 0.145 kg
  • Diameter d (m): 0.074 m
  • Drag coefficient C_d (0 = no air): 0.35
Answer
  • Range with drag: 93.9 m
  • Highest point with drag: 20.2 m
  • Flight time with drag: 4.05 s
  • Range without air: 153.3 m
  • Highest point without air: 26.8 m
Check
With C_d = 0 the result equals the textbook formula v₀² sin 2θ / g, and the drag case uses the same RK4 engine as the Physics Lab (Halliday ch. 4; Taylor ch. 2); the build checks both.

Assumptions

Sphere, sea-level air (1.225 kg/m³), constant C_d, no wind or spin, launch and landing at the same height.

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