A 63.0-kg bungee jumper is standing on a tall platform (h0 = 45.8 m). The bungee cord has an unstrained length of L0 = 9.18 m and, when stretched, behaves like an ideal spring with a spring constant of k = 67.2 N/m. The jumper falls from rest, and it is assumed that the only forces acting on him are his weight and, for the latter part of the descent, the elastic force of the bungee cord. Determine how far the bungee jumper is from the water when he reaches the lowest point in his fall.

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To determine how far the bungee jumper is from the water when he reaches the lowest point in his fall, we first need to calculate the distance he falls before the bungee cord starts to stretch. This can be calculated using the equation for the displacement of an object under constant acceleration:

Δy = (1/2) * a * t^2

where Δy is the displacement, a is the acceleration, and t is the time.

The acceleration can be found using Newton's second law of motion:

F = m * a

where F is the net force acting on the object and m is the mass. In this case, the net force is the gravitational force:

F = m * g

where g is the acceleration due to gravity. Substituting this into the equation, we get:

m * a = m * g

which means the acceleration is g.

Next, we can calculate the time it takes for the bungee cord to start stretching. This can be found using the equation:

h0 = (1/2) * g * t^2

Plugging in the given values, we can solve for t:

45.8 m = (1/2) * (9.8 m/s^2) * t^2

Solving this equation gives us t ≈ 3.01 seconds.

Now, we need to calculate the distance the bungee jumper falls while the cord is stretched. This can be found using Hooke's law, which states that the force exerted by a spring is proportional to the displacement from its equilibrium position:

F = -k * Δx

where F is the force, k is the spring constant, and Δx is the displacement from the equilibrium position. Rearranging this equation, we get:

Δx = -F / k

As the bungee cord is stretching, the force exerted by the cord is equal to the weight of the bungee jumper:

F = m * g

Substituting this into the equation, we get:

Δx = -(m * g) / k

Plugging in the given values, we can calculate Δx:

Δx = -(63.0 kg * 9.8 m/s^2) / 67.2 N/m ≈ -9.22 m

Note that the negative sign indicates that the displacement is in the opposite direction to the gravitational force.

Finally, we can calculate the total distance the bungee jumper falls by adding the distance fallen before the cord starts stretching to the distance fallen while the cord is stretched:

Total distance fallen = 45.8 m + (-9.22 m) = 36.58 m

Therefore, the bungee jumper is approximately 36.58 meters from the water when he reaches the lowest point in his fall.