[Leejohnston]

## Time Dilation — How Motion Changes Time

### 1. The Equation

Time dilation (special relativity):

Δt = Δt₀ ÷ √(1 − v² / c²)

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### 2. What Each Symbol Means

- Δt = observed time
- Δt₀ = proper time (time at rest)
- v = relative velocity
- c = speed of light

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### 3. What the Equation Is Telling Us

Moving clocks tick **more slowly** than stationary ones.

The faster you move, the slower your time passes relative to others.

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### 4. Where It Comes From (Intuition)

The speed of light must be the same for all observers.

Time adjusts to preserve that rule.

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### 5. Worked Example

If:
- v = 0.8c

Then:
√(1 − 0.64) = 0.6

Time runs at **60%** of normal.

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### 6. Real-World Applications

- GPS satellites
- Particle accelerators
- Space travel scenarios

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### 7. Common Misconceptions

- Time dilation is psychological → false
- Only noticeable near light speed → mostly true
- Time stops at c → undefined (cannot reach c)

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### Try It Yourself

How fast must you travel for time to slow by half?