A 61.0 -kg bungee jumper is standing on a tall platform (h0 = 46.0 m), as indicated in the figure. The bungee cord has an unstrained length of L0 = 9.00 m, and when stretched, behaves like an ideal spring with a spring constant of 76.0 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.

To determine how far the bungee jumper is from the water when he reaches the lowest point in his fall, we can use the principles of energy conservation.

Step 1: Find the potential energy of the jumper at the top of the fall.
The potential energy of an object is given by the equation PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height.
PE1 = m * g * h0

Step 2: Find the potential energy of the jumper at the lowest point.
Since the length of the bungee cord is stretched at the lowest point, the potential energy is given by the equation PE = (1/2) * k * x^2, where k is the spring constant and x is the displacement from the equilibrium position.
At the lowest point, the height is zero, so the potential energy is zero.
PE2 = 0

Step 3: Find the kinetic energy of the jumper at the lowest point.
Since the potential energy is converted into kinetic energy, the kinetic energy at the lowest point is equal to the potential energy at the top.
KE = PE1

Step 4: Find the displacement of the bungee jumper.
The displacement can be calculated using Hooke's law, which states that the force exerted by a spring is proportional to the displacement.
F = k * x
The gravitational force acting on the jumper is m * g.
At the lowest point, the spring force is equal to the gravitational force.
k * x = m * g
x = (m * g) / k

Substituting the values into the equation:
x = (61.0 kg * 9.8 m/s^2) / 76.0 N/m

Step 5: Calculate the displacement of the bungee jumper.
x = 7.85 m

Therefore, the bungee jumper is 7.85 m from the water when he reaches the lowest point in his fall.

To find the distance the bungee jumper is from the water when he reaches the lowest point in his fall, we first need to determine the total energy at the point where he starts falling, and then equate it to the total energy at the lowest point.

1. Calculate potential energy at starting point (PE_start):

The potential energy is given by the formula PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height from the starting point.

PE_start = m * g * h0

Substituting the given values, we get:

PE_start = 61.0 kg * 9.8 m/s^2 * 46.0 m

2. Calculate the potential energy when the bungee jumper is at the lowest point (PE_lowest):

At the lowest point, the bungee jumper has no potential energy, as he is at the lowest height. So we can set PE_lowest = 0.

3. Calculate the spring potential energy when the bungee jumper is at the lowest point (PE_spring):

The spring potential energy is given by the formula PE_spring = (1/2) * k * x^2, where k is the spring constant and x is the displacement from the equilibrium position.

At the lowest point, the spring is stretched by an additional distance equal to the unstrained length minus the displacement from the starting point, so x = L0 - h_lowest, where h_lowest is the position of the bungee jumper measured from the starting point.

PE_spring = (1/2) * k * (L0 - h_lowest)^2

4. Calculate the kinetic energy at the lowest point (KE_lowest):

At the lowest point, the bungee jumper has the maximum speed, and hence the maximum kinetic energy. So we can set KE_lowest = (1/2) * m * v^2, where v is the velocity at the lowest point.

Since the bungee jumper starts from rest, the initial velocity is zero, and the final velocity can be determined using the principle of conservation of mechanical energy (PE + KE = constant), i.e., PE_start + KE_start = PE_lowest + KE_lowest.

At the lowest point, the bungee jumper has zero potential energy (PE_lowest = 0).

KE_start = PE_start
(1/2) * m * v^2 = PE_start

Solving for v, we get:

v = sqrt(2 * PE_start / m)

Substituting the given values, we get:

v = sqrt(2 * (61.0 kg * 9.8 m/s^2 * 46.0 m) / 61.0 kg)

Now, substitute this value of v into KE_lowest to get the kinetic energy at the lowest point.

5. Solve for h_lowest:

Since the total energy (PE_lowest + PE_spring + KE_lowest) is conserved, we can set it equal to the total energy at the starting point (PE_start):

PE_start = PE_lowest + PE_spring + KE_lowest

Substitute the expressions for PE_start, PE_spring, and KE_lowest:

61.0 kg * 9.8 m/s^2 * 46.0 m = (1/2) * k * (L0 - h_lowest)^2 + (1/2) * m * v^2

Rearrange the equation and solve for h_lowest:

(1/2) * k * (L0 - h_lowest)^2 = 61.0 kg * 9.8 m/s^2 * 46.0 m - (1/2) * m * v^2

Once you have solved for h_lowest, the distance from the water when he reaches the lowest point is equal to h0 - h_lowest.