Projectile Motion Calculator
ANA›Life Services Authority›National Calculator Authority›Projectile Motion Calculator
.calc-container { max-width: 640px; margin: 2rem 0; padding: 1.5rem; background: #fff; border: 1px solid #ddd; border-radius: 8px; box-shadow: 0 1px 3px rgba(0,0,0,0.06); font-family: system-ui, -apple-system, sans-serif; } .calc-container h3 { font-family: Georgia, serif; font-size: 1.15rem; color: #1a1a1a; margin-bottom: 1rem; padding-bottom: 0.5rem; border-bottom: 2px solid var(--ac, #3d5a80); } .calc-row { display: flex; align-items: center; gap: 0.75rem; margin-bottom: 0.75rem; flex-wrap: wrap; } .calc-row label { min-width: 160px; font-size: 0.9rem; color: #333; font-weight: 500; } .calc-row input[type="number"], .calc-row select { flex: 1; min-width: 120px; max-width: 200px; padding: 0.5rem 0.6rem; border: 1px solid #ccc; border-radius: 4px; font-size: 0.9rem; font-family: system-ui, sans-serif; color: #1a1a1a; background: #fafaf8; } .calc-row input:focus, .calc-row select:focus { outline: none; border-color: var(--ac, #3d5a80); box-shadow: 0 0 0 2px rgba(26,74,138,0.12); } .calc-row .unit { font-size: 0.82rem; color: #888; min-width: 30px; } .calc-btn { display: inline-block; margin-top: 0.5rem; padding: 0.55rem 1.5rem; background: var(--ac, #3d5a80); color: #fff; border: none; border-radius: 4px; font-size: 0.9rem; font-weight: 600; cursor: pointer; font-family: system-ui, sans-serif; } .calc-btn:hover { opacity: 0.9; } .calc-result { margin-top: 1.25rem; padding: 1rem 1.25rem; background: #f0f6fc; border-left: 3px solid var(--ac, #3d5a80); border-radius: 0 6px 6px 0; display: none; } .calc-result.visible { display: block; } .calc-result-label { font-size: 0.78rem; text-transform: uppercase; letter-spacing: 0.06em; color: #666; margin-bottom: 0.25rem; } .calc-result-value { font-size: 1.6rem; font-weight: 700; color: var(--ac, #3d5a80); } .calc-result-detail { font-size: 0.85rem; color: #555; margin-top: 0.5rem; line-height: 1.5; } .calc-note { margin-top: 1rem; font-size: 0.8rem; color: #888; font-style: italic; } .calc-grid { display: grid; grid-template-columns: 1fr 1fr; gap: 0.75rem; margin-top: 0.75rem; } .calc-grid-item { padding: 0.6rem 0.8rem; background: #f8f9fa; border-radius: 4px; border: 1px solid #eee; } .calc-grid-item .label { font-size: 0.75rem; color: #888; text-transform: uppercase; letter-spacing: 0.04em; } .calc-grid-item .value { font-size: 1.1rem; font-weight: 600; color: #1a1a1a; } @media (max-width: 720px) { .calc-row { flex-direction: column; align-items: flex-start; gap: 0.3rem; } .calc-row label { min-width: auto; } .calc-row input[type="number"], .calc-row select { max-width: 100%; width: 100%; } .calc-grid { grid-template-columns: 1fr; } } .calc-chart { margin: 1rem 0; text-align: center; } .calc-chart svg { max-width: 100%; height: auto; } .calc-chart-legend { display: flex; flex-wrap: wrap; justify-content: center; gap: 0.6rem 1.2rem; margin-top: 0.6rem; font-size: 0.8rem; color: #555; } .calc-chart-legend span { display: inline-flex; align-items: center; gap: 0.3rem; } .calc-chart-legend i { display: inline-block; width: 10px; height: 10px; border-radius: 2px; font-style: normal; } .calc-related { max-width: 640px; margin: 2rem 0 1rem; padding: 1.25rem 1.5rem; background: #f8f9fa; border: 1px solid #e8e8e8; border-radius: 8px; } .calc-related h3 { font-family: Georgia, serif; font-size: 1rem; color: #1a1a1a; margin: 0 0 0.75rem; padding-bottom: 0.4rem; border-bottom: 2px solid var(--ac, #3d5a80); } .calc-related-list { list-style: none; padding: 0; margin: 0 0 0.75rem; display: grid; grid-template-columns: 1fr 1fr; gap: 0.4rem 1.5rem; } .calc-related-list li a { font-size: 0.88rem; color: var(--ac, #3d5a80); text-decoration: none; } .calc-related-list li a:hover { text-decoration: underline; } .calc-browse-all { margin: 0.5rem 0 0; font-size: 0.9rem; font-weight: 600; } .calc-browse-all a { color: var(--ac, #3d5a80); text-decoration: none; } .calc-browse-all a:hover { text-decoration: underline; } @media (max-width: 720px) { .calc-related-list { grid-template-columns: 1fr; } }
Projectile Motion Calculator
Calculate range, maximum height, time of flight, and final velocity of a projectile given initial velocity, launch angle, and optional initial height.
Initial Velocity (m/s)
Launch Angle (degrees, 0–90)
Initial Height (m, default 0)
Gravitational Acceleration (m/s², default 9.81)
Calculate Results will appear here.
function proCalc() { const resultDiv = document.getElementById('pro-result');
const v0 = parseFloat(document.getElementById('pro-velocity').value); const angleDeg = parseFloat(document.getElementById('pro-angle').value); const h0 = parseFloat(document.getElementById('pro-height').value) || 0; const g = parseFloat(document.getElementById('pro-gravity').value) || 9.81;
// --- Validation --- if (isNaN(v0) || isNaN(angleDeg)) { resultDiv.innerHTML = 'Please enter Initial Velocity and Launch Angle.'; return; } if (v0 Initial velocity must be non-negative.'; return; } if (angleDeg 90) { resultDiv.innerHTML = 'Launch angle must be between 0° and 90°.'; return; } if (h0 Initial height must be non-negative.'; return; } if (g Gravitational acceleration must be positive.'; return; }
// --- Core Calculations --- const angleRad = angleDeg * Math.PI / 180; const v0x = v0 * Math.cos(angleRad); // Horizontal component const v0y = v0 * Math.sin(angleRad); // Vertical component
// Time of flight: solve 0 = h0 + v0yt - 0.5gt² // 0.5gt² - v0yt - h0 = 0 → quadratic in t // t = (v0y + sqrt(v0y² + 2gh0)) / g (positive root) const discriminant = v0y * v0y + 2 * g * h0; const tFlight = (v0y + Math.sqrt(discriminant)) / g;
// Horizontal range const range = v0x * tFlight;
// Maximum height: h_max = h0 + v0y²/(2g) const hMax = h0 + (v0y * v0y) / (2 * g);
// Time to reach max height const tMax = v0y / g;
// Final velocity components at impact const vxFinal = v0x; // Horizontal velocity is constant const vyFinal = v0y - g * tFlight; // Vertical velocity at impact (negative = downward) const vFinal = Math.sqrt(vxFinal * vxFinal + vyFinal * vyFinal);
// Impact angle below horizontal const impactAngle = Math.atan2(Math.abs(vyFinal), Math.abs(vxFinal)) * 180 / Math.PI;
// --- Display Results --- resultDiv.innerHTML = 'Results:' + 'Horizontal Velocity (v₀ₓ): ' + v0x.toFixed(4) + ' m/s' + 'Vertical Velocity (v₀ᵧ): ' + v0y.toFixed(4) + ' m/s' + 'Time of Flight: ' + tFlight.toFixed(4) + ' s' + 'Time to Max Height: ' + tMax.toFixed(4) + ' s' + 'Maximum Height: ' + hMax.toFixed(4) + ' m' + 'Horizontal Range: ' + range.toFixed(4) + ' m' + 'Final Speed at Impact: ' + vFinal.toFixed(4) + ' m/s' + 'Impact Angle Below Horizontal: ' + impactAngle.toFixed(4) + '°'; }
#### Formulas Used
Velocity Components: v₀ₓ = v₀ · cos(θ) v₀ᵧ = v₀ · sin(θ)
Time of Flight (from h₀ to ground, h = 0): 0 = h₀ + v₀ᵧ·t − ½·g·t² t = (v₀ᵧ + √(v₀ᵧ² + 2·g·h₀)) / g
Horizontal Range: R = v₀ₓ · t
Maximum Height: H_max = h₀ + v₀ᵧ² / (2·g)
Time to Maximum Height: t_max = v₀ᵧ / g
Final Speed at Impact: v_f = √(v₀ₓ² + vᵧ_f²), where vᵧ_f = v₀ᵧ − g·t
Impact Angle Below Horizontal: θ_impact = arctan(|vᵧ_f| / |v₀ₓ|)
#### Assumptions & References
- Reference: Halliday, Resnick & Krane, Physics, 5th ed., Chapter 4.
- Reference: Serway & Jewett, Physics for Scientists and Engineers, 8th ed.
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