---
title: "Half-life — AP Physics 2 Definition & Exam Guide"
description: "Half-life is the time for half of a radioactive sample to decay. Learn how it ties to decay probability, the decay constant, and Unit 7 of AP Physics 2."
canonical: "https://fiveable.me/ap-physics-2-revised/key-terms/half-life"
type: "key-term"
subject: "AP Physics 2"
unit: "Unit 7"
---

# Half-life — AP Physics 2 Definition & Exam Guide

## Definition

Half-life is the time it takes for half of the radioactive nuclei in a sample to decay. In AP Physics 2, it shows up in Unit 7 (Topic 7.7) as the clearest real-world example of quantum probability, since you can never predict when one nucleus decays, only the odds.

## What It Is

Half-life is the time required for half of the [radioactive nuclei](/ap-physics-2-revised/unit-15/7-fission-fusion-and-nuclear-decay/study-guide/nVULWHgdGjkiGhbI "fv-autolink") in a sample to undergo [radioactive decay](/ap-physics-2-revised/key-terms/radioactive-decay "fv-autolink"). Start with 1,000 unstable nuclei, wait one half-life, and on average about 500 remain. Wait another half-life and you're down to about 250. The amount remaining follows the pattern N = N₀(1/2)^(t/T½), so the sample never hits exactly zero. It just keeps halving.

Here's the part [AP Physics 2](/ap-physics-2-revised "fv-autolink") actually cares about. Radioactive decay is fundamentally random. You cannot point at one specific nucleus and say when it will decay. All you know is its probability of decaying in a given time interval. Half-life is what that probability looks like when you zoom out to a huge number of nuclei. It's the same logic as flipping a billion coins: any single flip is unpredictable, but the overall behavior is rock-solid. That's why half-life lives in Topic 7.7, Wave Functions and Probability. It's quantum randomness producing a predictable macroscopic rule.

## Why It Matters

Half-life sits in Unit 7 (Quantum, Atomic, and Nuclear Physics) under Topic 7.7, [Wave](/ap-physics-2-revised/unit-14/1-properties-of-wave-pulses-and-waves/study-guide/Ql0FLnrI6dIHcNlL "fv-autolink") Functions and Probability. The big idea in that topic is that quantum mechanics only gives you probabilities, and half-life is the most concrete, testable example of that idea. It also connects forward to nuclear processes like fission and radioactive dating, where the exponential decay pattern does the heavy lifting. Conceptually, half-life trains you to think the way the exam wants: reason about a system statistically instead of tracking individual particles. If a question asks why we can't say when a single [nucleus](/ap-physics-2-revised/key-terms/nucleus "fv-autolink") decays but can say a sample's activity drops by half every 5,730 years, the answer is the probabilistic nature of quantum systems, and half-life is the bridge between those two statements.

## Connections

### Decay constant (Unit 7)

The decay constant λ is the per-nucleus probability of decaying per unit time, and half-life is its flip side. They're tied together by T½ = ln(2)/λ. A big decay constant means a short half-life, because nuclei that are more likely to decay each second don't stick around long.

### Radioactive dating (Unit 7)

Dating is half-life run in reverse. Measure how much of an [isotope](/ap-physics-2-revised/key-terms/isotope "fv-autolink") remains, count how many halvings that represents, and multiply by the half-life to get the age. Carbon-14 dating is the classic example.

### Wave functions and probability (Topic 7.7)

Half-life is statistical, not deterministic, for the same reason a wave function only gives the probability of finding a particle somewhere. Quantum mechanics predicts odds, not outcomes, and decay is where that randomness becomes something you can measure with a Geiger counter.

### [Mass-Energy Equivalence (Unit 7)](/ap-physics-2-revised/key-terms/mass-energy-equivalence)

Half-life tells you when decay happens; E = mc² tells you what you get. Each decay converts a tiny mass difference between parent and daughter nuclei into the [kinetic energy](/ap-physics-2-revised/key-terms/kinetic-energy "fv-autolink") of the emitted particles.

## On the AP Exam

Half-life questions in AP Physics 2 are usually conceptual or semi-quantitative, not heavy calculation. Expect MCQ stems like "after three half-lives, what fraction of the original sample remains?" (answer: 1/8, since each half-life cuts the sample in half again) or graph-reading questions where you identify the half-life from an N vs. t decay curve by finding the time to drop to 50%. The other angle is probabilistic reasoning. A question might ask why two identical nuclei can decay at very different times, and the credited response invokes the random, probability-governed nature of quantum processes. No released FRQ has hinged on the term verbatim, but the underlying skill, reading exponential decay behavior and explaining quantum randomness, fits directly into Unit 7 conceptual questions. Be ready to count halvings, read decay graphs, and write a sentence connecting decay to probability.

## Half-life vs Decay constant

Half-life and the decay constant describe the same physics from opposite directions. Half-life (T½) is a time, how long until half the sample is gone. The decay constant (λ) is a rate, the probability per second that any one nucleus decays. They're inversely related through T½ = ln(2)/λ, so a short half-life means a large decay constant. Mixing them up flips your answer: a "highly radioactive" isotope has a HIGH decay constant but a SHORT half-life.

## Key Takeaways

- Half-life is the time for half of a radioactive sample to decay, so after n half-lives the fraction remaining is (1/2)^n.
- Decay is probabilistic. You can never predict when one specific nucleus decays, only the odds, which is why half-life belongs in Topic 7.7 with wave functions and probability.
- Half-life and the decay constant are inverses in spirit: T½ = ln(2)/λ, so more probable decay per second means a shorter half-life.
- On a decay graph, the half-life is the time it takes the curve to fall to 50% of its starting value, and that time is the same no matter where on the curve you start.
- The sample never reaches exactly zero. Exponential decay keeps halving forever, which is what makes radioactive dating possible.

## FAQs

### What is half-life in AP Physics 2?

Half-life is the time it takes for half of the radioactive nuclei in a sample to decay. It appears in Unit 7 (Topic 7.7) as an example of quantum probability, since individual decays are random but large samples follow a predictable exponential pattern.

### Does a sample completely disappear after two half-lives?

No. After two half-lives, 25% of the original nuclei remain, not 0%. Each half-life cuts the remaining amount in half, so the sample approaches zero but never reaches it. This is a classic MCQ trap.

### What's the difference between half-life and decay constant?

Half-life is a time (how long until half the sample is gone) while the decay constant is a probability per unit time for a single nucleus. They're linked by T½ = ln(2)/λ, so a larger decay constant means a shorter half-life.

### Can you predict when a single nucleus will decay?

No. Quantum mechanics only gives the probability of decay in a given time interval, never the exact moment. Half-life is a statistical statement about large numbers of nuclei, which is exactly the probabilistic thinking Topic 7.7 tests.

### What fraction of a sample is left after 3 half-lives?

One eighth (12.5%). Each half-life halves the sample, so the math is (1/2)³ = 1/8. Counting halvings like this is the most common half-life calculation on the exam.

## Related Study Guides

- [7.7 Wave Functions and Probability](/ap-physics-2-revised/unit-7/wave-functions-probability/study-guide/HYPoYs3xG1I9JCLLP09U)
- [Unit 7 Overview: Quantum, Atomic, and Nuclear Physics](/ap-physics-2-revised/unit-QUWmBhy0eEwkfCSU/review/study-guide/rcA6NJTYu2BTlTMRS2dT)

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