Calculate remaining value, mass, or quantity over time.
Discrete Decay (e.g., Depreciation: A = P(1 – r)^t)
Continuous Decay (e.g., Radioactive: N = N₀e^(-kt))
Best for assets that lose value annually or periodically.
Initial Quantity:–
Decay Rate:–
Time Elapsed:–
Total Amount Lost:–
Remaining Quantity:–
Estimated Half-Life:–
// Variable declaration using var as requested
var modelSelect = document.getElementById('decayModel');
var modelNote = document.getElementById('modelNote');
var rateLabel = document.getElementById('rateLabel');
function updateLabels() {
var model = modelSelect.value;
if (model === 'discrete') {
modelNote.innerText = "Best for assets that lose value annually or periodically (e.g., car value).";
rateLabel.innerText = "Decay Rate per Period (%)";
} else {
modelNote.innerText = "Best for natural processes like radioactive decay or bacterial cooling.";
rateLabel.innerText = "Decay Constant (k) or Rate (%)";
}
}
function calculateDecay() {
var initial = parseFloat(document.getElementById('initialValue').value);
var ratePercent = parseFloat(document.getElementById('decayRate').value);
var time = parseFloat(document.getElementById('timeElapsed').value);
var model = document.getElementById('decayModel').value;
var resultArea = document.getElementById('result-area');
// Error handling
if (isNaN(initial) || isNaN(ratePercent) || isNaN(time)) {
alert("Please enter valid numeric values for all fields.");
return;
}
if (initial < 0 || time 1) {
alert("For discrete decay, the rate cannot exceed 100%.");
return;
}
finalValue = initial * Math.pow((1 – rateDecimal), time);
document.getElementById('halfLifeRow').style.display = 'none';
} else {
// Formula: N = N0 * e^(-kt)
// Here rateDecimal is treated as k (lambda)
finalValue = initial * Math.exp(-rateDecimal * time);
// Calculate Half-Life: t(1/2) = ln(2) / k
if (rateDecimal > 0) {
halfLife = Math.log(2) / rateDecimal;
document.getElementById('displayHalfLife').innerText = halfLife.toFixed(4) + " time units";
document.getElementById('halfLifeRow').style.display = 'flex';
} else {
document.getElementById('halfLifeRow').style.display = 'none';
}
}
var amountLost = initial – finalValue;
// Display Results
document.getElementById('displayInitial').innerText = initial.toLocaleString(undefined, {maximumFractionDigits: 2});
document.getElementById('displayRate').innerText = ratePercent + "%";
document.getElementById('displayTime').innerText = time;
document.getElementById('displayFinal').innerText = finalValue.toLocaleString(undefined, {maximumFractionDigits: 4});
document.getElementById('displayLost').innerText = amountLost.toLocaleString(undefined, {maximumFractionDigits: 4});
resultArea.style.display = 'block';
}
Understanding the Decay Rate Formula
Exponential decay is a process where a quantity decreases at a rate proportional to its current value. This concept is fundamental in various fields, including physics (radioactive decay), finance (depreciation), biology (pharmacokinetics), and environmental science.
The Two Main Types of Decay
Depending on the context, exponential decay is calculated using two slightly different models. This calculator supports both:
1. Discrete Decay (Depreciation)
This model is used when decay happens at specific intervals, such as a car losing value at the end of every year.
A = P(1 – r)t
A: The final amount remaining.
P: The initial principal amount.
r: The rate of decay per period (expressed as a decimal).
t: The number of time periods elapsed.
2. Continuous Decay (Exponential)
This model is used for natural processes that happen continuously, such as the cooling of a hot object or the decay of radioactive isotopes.
N(t) = N₀e-kt
N(t): The quantity at time t.
N₀: The initial quantity.
e: Euler's number (approx. 2.71828).
k: The continuous decay constant (or rate).
t: Time elapsed.
Real-World Examples
Example 1: Car Depreciation (Discrete)
If you buy a car for 25,000 units of currency and it depreciates at a rate of 15% per year, what is it worth after 5 years?
Using the discrete formula: 25,000 * (1 – 0.15)5 ≈ 11,092.63.
Example 2: Radioactive Isotope (Continuous)
A sample contains 100g of a substance with a decay constant of 0.05 (5%). How much remains after 10 hours?
Using the continuous formula: 100 * e-(0.05 * 10) = 100 * e-0.5 ≈ 60.65g.
What is Half-Life?
In continuous decay, the half-life is the time required for a quantity to reduce to half of its initial value. It is related to the decay constant ($k$) by the formula:
t1/2 = ln(2) / k ≈ 0.693 / k
This calculator automatically estimates the half-life when using the Continuous Decay mode.