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Dynamics Calculator — Projectiles & Vibration

Two exam-workhorse calculators in one place: projectile trajectory (range, maximum height, time of flight) and single-degree-of-freedom free vibration (natural frequency, period, damped frequency). Pick your unit system, enter the inputs, and check the result against your hand calculation.

Projectile motion

R = v₀² sin 2θ/g h_max = v₀² sin²θ/(2g) T = 2v₀ sin θ/g

v₀	launch speed (m/s or ft/s)
θ	launch angle above the horizontal (degrees)
g	9.81 m/s² (SI) or 32.2 ft/s² (US)

Assumes launch and landing at the same elevation and no air resistance — the standard FE idealisation.

Unit system Launch speed v₀ (m/s) Launch angle θ (degrees)

Mass–spring vibration

ω_n = √(k/m) f_n = ω_n/(2π) τ = 1/f_n f_d = f_n√(1 − ζ²)

m	mass: enter kg in SI; in US units enter the weight in lb and the calculator converts with m = W/g
k	spring stiffness (N/m or lb/ft)
ζ	damping ratio, dimensionless (0 for undamped)

Unit system Mass (kg) Stiffness k (N/m) Damping ratio ζ

For study and checking only. These calculators run entirely in your browser — nothing is sent anywhere. They are meant to verify your hand calculations while you study; on the exam you will work by hand with the NCEES reference handbook.

Worked example: projectile range, height, and flight time

Launch speed v₀ = 25 m/s at θ = 35° above the horizontal, from ground level (g = 9.81 m/s²). Find R, h_max, and T.

v_0x = 25 cos 35° = 20.5 m/s; v_0y = 25 sin 35° = 14.3 m/s
T = 2 × 14.3/9.81 = 2.92 s; R = 20.5 × 2.92 = 59.9 m; h_max = 14.3²/(2 × 9.81) = 10.5 m
Cross-check: R = 25² sin 70°/9.81 = 59.9 m ✓

Exam tip: range needs sin 2θ — at 45° it doesn’t matter (sin 90° = 1), so the trap only bites at every other launch angle.

Go deeper

Preparing for exam day? The 110-question FE Civil practice exam is a full 5-hour-20-minute rehearsal with detailed solutions.