# Homework Help: Gravitational Time Dilation Problem

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1. Feb 3, 2016

### J_M_R

1. The problem statement, all variables and given/known data

A person lived 75 years in a city located 3.1km above sea level. How much longer could they have lived at sea level? (Times are measured by an observer at infinite distance).

2. Relevant equations

tr/t∞ = {1 - [ (2GM) / (r(c^2)) ]}^(1/2)

Rc (Radius at city) = Rearth + 3.1km

∴ t(sea-level)/t∞ = {1 - [ (2GMe) / (Rearth(c^2)) ]}^(1/2)

and ∴ t(city)/t∞ = {1 - [ (2GMe) / (Rc(c^2)) ]}^(1/2)

3. The attempt at a solution

t(sea-level) / {1 - [ (2GMe) / (Rearth(c^2)) ]}^(1/2) ≈ t(city) / {1 - [ (2GMe) / (Rc(c^2)) ]}^(1/2)

Having made the the two t∞ equal to each other.

Knowing t(city) = 75 years this gave t(sea-level) as 74.99 years.

Where have I gone wrong as shouldn't t(sea-level) be longer?

2. Feb 3, 2016

### PeroK

What gives you that idea?

3. Feb 5, 2016

### J_M_R

I was just using the fact that the question says "How much longer could they have lived at sea level?" so assumed t(sea-level) should therefore be longer?

4. Feb 5, 2016

### Bandersnatch

Ask yourself, which of the two has longer until his 76th birthday according to your calculations.

5. Feb 5, 2016

### J_M_R

Ah, so the t(sea-level) has longer until his 76th birthday so the person could have lived 0.01 years longer at sea level according to my calculations?

6. Feb 5, 2016

You got it.

7. Feb 5, 2016

Thanks!