A 82.5-kg climber is scaling the vertical wall of a mountain. His safety rope is made of a material that, when stretched, behaves like a spring with a spring constant of 1.82 x 103 N/m. He accidentally slips and falls freely for 0.739 m before the rope runs out of slack. How much is the rope stretched when it breaks his fall and momentarily brings him to rest?

To solve this problem, we can use the principle of conservation of mechanical energy. Initially, the climber has gravitational potential energy due to his height, and when the rope reaches its maximum stretch, he has elastic potential energy.

Let's denote the following variables:
- Mass of the climber (m) = 82.5 kg
- Free-fall distance (d) = 0.739 m
- Spring constant of the rope (k) = 1.82 x 10^3 N/m

1. First, let's calculate the gravitational potential energy of the climber as he falls freely:
Potential Energy (PE) = mass (m) × gravitational acceleration (g) × height (h)

Since the height (h) is equal to the free-fall distance (d), and the gravitational acceleration (g) is approximately 9.8 m/s², we have:
PE = m × g × d

2. Next, let's calculate the elastic potential energy at the maximum stretch of the rope:
Elastic Potential Energy (PE) = 0.5 × k × x²

Here, x represents the amount by which the rope stretches. At the maximum stretch, the climber momentarily comes to rest, so his kinetic energy is zero.

3. According to the conservation of mechanical energy:
Initial Energy = Final Energy

Since the climber's initial energy is only gravitational potential energy (PE), and his final energy is only elastic potential energy (PE), we have:
m × g × d = 0.5 × k × x²

Now, we can substitute the given values into the equation and solve for x:

m × g × d = 0.5 × k × x²

Plugging in the values:
x² = (2 × m × g × d) / k

Calculating:
x² = (2 × 82.5 kg × 9.8 m/s² × 0.739 m) / (1.82 × 10^3 N/m)
x² ≈ 6.164

Taking the square root of both sides:
x ≈ √6.164
x ≈ 2.482 m

Therefore, the rope stretches approximately 2.482 meters when it breaks the climber's fall and momentarily brings him to rest.