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Category Physics & Science

Launch Parameters Angle 45° Speed 25 m/s Launch Height 1.0 m

Environment Gravity 9.81 m/s² Air Density 1.225 kg/m³ Wind 0 m/s

Ball Properties Mass 0.145 kg Radius 0.037 m Drag Coeff. 0.47 Restitution 0.70

Display Show vacuum trajectory Show trail

Presets

Range —

Max Height —

Flight Time —

Impact Speed —

Bounces —

Vacuum Range —

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About

Projectile motion calculations without air resistance are trivial. Real trajectories diverge significantly from the parabolic ideal due to quadratic drag force proportional to v². A baseball launched at 45° with initial speed 40 m/s lands roughly 30% shorter than the vacuum prediction. This simulator uses Euler numerical integration to solve the coupled differential equations of motion frame-by-frame, applying gravitational acceleration g, quadratic aerodynamic drag with coefficient C_d, and optional constant wind force. Bounce collisions apply a coefficient of restitution e that removes kinetic energy each impact. The tool assumes a spherical projectile, sea-level air density ρ = 1.225 kg/m³, and no spin (Magnus effect is excluded). Approximation error scales with timestep size. Results are visual estimates, not engineering-grade predictions.

Formulas

The net force on the projectile is the vector sum of gravity, aerodynamic drag, and wind. At each timestep Δt, position and velocity are updated via forward Euler integration.

F = mg − 12 C_d ρ A |v|² v

Where the cross-sectional area is:

A = π r²

Velocity update per timestep:

v_n+1 = v_n + Fm Δt

Position update:

x_n+1 = x_n + v_n+1 Δt

On ground collision (y ≤ 0), the vertical velocity component is reversed and scaled by the coefficient of restitution:

v_y′ = −e ⋅ v_y

Where m = mass, C_d = drag coefficient, ρ = air density, A = cross-sectional area, r = ball radius, e = coefficient of restitution, v = unit velocity vector. Wind adds a constant horizontal acceleration a_w independent of ball velocity.

Reference Data

Parameter	Symbol	Default	Typical Range	Notes
Gravitational Acceleration	g	9.81 m/s²	1.62 - 24.79	Moon: 1.62, Jupiter: 24.79
Drag Coefficient (Sphere)	C_d	0.47	0.1 - 1.2	Smooth sphere at Re > 10³
Air Density (Sea Level)	ρ	1.225 kg/m³	0.0 - 1.5	Set 0 for vacuum
Ball Mass	m	0.145 kg	0.01 - 10	Baseball: 0.145, Soccer: 0.43
Ball Radius	r	0.037 m	0.01 - 0.5	Cross-section A = πr²
Coefficient of Restitution	e	0.7	0.0 - 1.0	1.0 = perfectly elastic
Launch Angle	θ	45°	0 - 90	Optimal vacuum range at 45°
Launch Speed	v₀	25 m/s	1 - 100	MLB pitch: ~40 m/s
Launch Height	h₀	1 m	0 - 50	Height above ground plane
Wind Speed	v_w	0 m/s	−30 - 30	Positive = tailwind, Negative = headwind
Timestep	Δt	0.002 s	0.001 - 0.01	Smaller = more accurate, slower
Tennis Ball C_d	-	0.55	-	Fuzzy surface increases drag
Golf Ball C_d	-	0.25	-	Dimples reduce drag via turbulent boundary layer
Basketball C_d	-	0.47	-	Standard smooth sphere approximation
Ping Pong Ball	-	0.50	-	Mass: 0.0027 kg, very drag-sensitive
Mars Gravity	g	3.72 m/s²	-	Thin atmosphere: ρ ≈ 0.020

Frequently Asked Questions

The parabolic trajectory assumes zero air resistance. In reality, drag force scales with the square of velocity (F_drag ∝ v²), which decelerates the ball significantly. At launch speed of 40 m/s with C_d = 0.47, drag removes roughly 25-35% of the range compared to vacuum. Set air density ρ to 0 to see the ideal parabola for comparison.

Heavier balls are less affected by drag because the drag-to-weight ratio decreases. A ping pong ball (0.0027 kg) is extremely drag-sensitive and barely travels 10 m at 25 m/s, while a baseball (0.145 kg) at the same speed reaches over 50 m. This is why artillery shells, despite their size, maintain range better than tennis balls.

Typical values: tennis ball on hard court ≈ 0.75, basketball on hardwood ≈ 0.80, golf ball on concrete ≈ 0.85, steel ball on steel ≈ 0.95, clay ball (dead bounce) ≈ 0.15-0.30. A value of 1.0 means perfectly elastic (no energy loss). Values above 0.9 produce long bounce sequences; below 0.3 the ball stops quickly.

In vacuum, 45° maximizes range because the horizontal and vertical velocity components contribute equally to the parabolic path. With drag, higher trajectories mean longer flight times and more cumulative deceleration. The optimal angle drops to approximately 35-42° depending on drag intensity. For highly drag-sensitive projectiles like shuttlecocks, optimal angles can be as low as 20-25°.

Forward Euler is a first-order method with local truncation error proportional to Δt². At the default timestep of 0.002 s, positional error accumulates to roughly 0.1-0.5% over a typical 3-second flight. For visual simulation this is sufficient. For engineering applications, fourth-order Runge-Kutta (RK4) would be preferred. You can observe integration accuracy by reducing the timestep and comparing results.

Yes. Set gravitational acceleration to the planet's surface gravity: Moon = 1.62 m/s², Mars = 3.72 m/s², Venus = 8.87 m/s², Jupiter = 24.79 m/s². Also adjust air density: Moon and Mercury have effectively 0 kg/m³, Mars ≈ 0.020 kg/m³, Venus ≈ 65 kg/m³ (extremely dense). On Venus, drag dominates completely even at low speeds.