Rate of Change of Angle of Elevation Calculator

Calculate how fast an observer's viewing angle changes for a moving object.

Altitude (h) Vertical height of the object (meters or feet)

Horizontal Distance (x) Distance along the ground from observer

Horizontal Velocity (v) Speed of the object moving horizontally (m/s or ft/s)

Calculation Results:

Current Angle of Elevation (θ):

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Rate of Change (Radians/sec):

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Rate of Change (Degrees/sec):

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Understanding Rate of Change of Elevation

In calculus, the rate of change of the angle of elevation is a related rates problem. It measures how quickly the viewing angle (θ) between an observer on the ground and a moving object in the air changes over time (dθ/dt).

The Physics Formula

Assuming an object is moving horizontally at a constant altitude h with a constant velocity v, the relationship is defined by:

            dθ/dt = (v · h) / (x² + h²)
        

v: Horizontal velocity of the object.
h: Constant altitude.
x: Horizontal distance from the observer.

Practical Example

Imagine a plane flying at a constant altitude of 3,000 meters at a speed of 200 meters per second. If the plane is currently 4,000 meters away (horizontally) from you, how fast must you tilt your head back to keep watching it?

Altitude (h) = 3,000 m
Distance (x) = 4,000 m
Velocity (v) = 200 m/s
Calculation: (200 * 3000) / (4000² + 3000²) = 600,000 / 25,000,000 = 0.024 rad/sec.
Converting to degrees: 0.024 * (180/π) ≈ 1.375° per second.

function calculateRateOfChange() { var h = parseFloat(document.getElementById('altitude_h').value); var x = parseFloat(document.getElementById('distance_x').value); var v = parseFloat(document.getElementById('velocity_v').value); if (isNaN(h) || isNaN(x) || isNaN(v)) { alert("Please enter valid numeric values for all fields."); return; } if (h theta = atan(h/x) var thetaRad = Math.atan2(h, x); var thetaDeg = thetaRad * (180 / Math.PI); // Step 2: Calculate d(theta)/dt // Formula: d(theta)/dt = (v * h) / (x^2 + h^2) // Note: This assumes v is moving toward the observer if x is decreasing, // but typically we express the magnitude of the rate. var denominator = (x * x) + (h * h); var dThetaDtRad = (v * h) / denominator; // Step 3: Convert to degrees/sec var dThetaDtDeg = dThetaDtRad * (180 / Math.PI); // Display results document.getElementById('res_angle').innerText = thetaDeg.toFixed(2) + "°"; document.getElementById('res_rads').innerText = dThetaDtRad.toFixed(5) + " rad/s"; document.getElementById('res_degs').innerText = dThetaDtDeg.toFixed(3) + "°/s"; document.getElementById('results_area').style.display = 'block'; }