Projectile motion with air drag

Inputs & what-if

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Projectile Motion (point mass)

Classical mechanics

The scenario involves a body leaving a ramp and travelling through the air under gravity, so a ballistic trajectory with optional aerodynamic drag is the appropriate model.

Equations

• v₀ₓ = v₀·cos(θ), v₀ᵧ = v₀·sin(θ)
• a = −g ŷ − (½ρ·CdA/m)·|v|·v
• x(t+Δt) = x(t) + vₓΔt (Δt = 2 ms, semi-implicit Euler)
• E_k = ½mv², p = mv

Validity range

Speeds 5–200 km/h, angles 0–60°, sub-transonic, near-sea-level air density.

Limitations

• This is a simplified physics model, not a full multi-body vehicle dynamics simulation.
• Pitch rotation is illustrative only; real rotation depends on suspension rebound and CG position.
• Landing damage, rollover and chassis loads are not computed.

Sensitivity

Effect of a +10% change on Air time

Timeline

• 0.00 sTakeoff — object leaves the ramp
• 1.40 sApex reached at 13.71 m
• 3.24 sLanding at 34.56 m from takeoff

External data & sources

• Regulation baseball massofficial · high

  142–149 g

  MLB official rules, ball specification · retrieved 1970-01-01

Assumptions

• Still air
• No spin-induced lift (Magnus effect ignored)
• Vehicle treated as a point mass at its centre of gravity.
• Ramp exit velocity equals approach speed (no traction/rolling losses on the ramp).
• Flat, level landing surface at ramp-exit height reference.
• No suspension travel, tyre deformation or aerodynamic lift.

Results

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Scientific integrity

Results are computed by an explicit, deterministic model integrated in SI units. Values are labelled as user input, external data, assumption, estimate or calculated result. Simplified models are educational approximations, not engineering-grade predictions.