A 100 kg. man decides to go bungee jumping off a bridge. The bungee cord has a relaxed length of 10.0 m, and acts as a perfect spring with a stiffness constant of 80 N/m. What is the maximum distance below the bridge that he will reach? (he will fall 10.0 m. before the bungee cord even starts to stretch)

change in potential energy due to fall

= m g (10+x)

storage of potential energy in spring = (1/2) k x^2
so

(1/2) k x^2 = m g (10 + x)

you have everything, solve quadratic
for x then:
distance below bridge = 10 + x

To find the maximum distance below the bridge, we need to calculate the extension of the bungee cord when it reaches its maximum stretch. This can be done using Hooke's Law, which states that the force exerted by a spring is directly proportional to the extension of the spring from its equilibrium position.

Here are the steps to calculate the maximum distance below the bridge:

Step 1: Calculate the force acting on the bungee cord
The force acting on the bungee cord is the weight of the man. Since the mass of the man is 100 kg and the acceleration due to gravity is 9.8 m/s², the force is given by:

Force = mass x acceleration due to gravity
Force = 100 kg x 9.8 m/s²
Force = 980 N

Step 2: Calculate the extension of the bungee cord
The stiffness constant of the bungee cord is given as 80 N/m. The extension of the bungee cord is given by:

Extension = Force / Stiffness constant
Extension = 980 N / 80 N/m
Extension = 12.25 m

Step 3: Calculate the maximum distance below the bridge
Since the bungee cord already falls for 10.0 m before it starts stretching, we subtract this initial fall distance from the extension calculated in Step 2:

Maximum distance below the bridge = Extension - Initial fall distance
Maximum distance below the bridge = 12.25 m - 10.0 m
Maximum distance below the bridge = 2.25 m

Therefore, the maximum distance below the bridge that the man will reach is 2.25 meters.

To determine the maximum distance below the bridge that the man will reach during his bungee jump, we can use the concepts of potential energy and the principle of conservation of mechanical energy.

Here are the steps to calculate the maximum distance:

Step 1: Calculate the initial potential energy (before the bungee cord starts stretching):
The initial potential energy is given by the formula:
Potential Energy = mass * gravity * height
= 100 kg * 9.8 m/s^2 * 10.0 m
= 9800 J

Step 2: Calculate the total potential energy when the cord is fully stretched:
The total potential energy is the sum of the initial potential energy and the elastic potential energy stored in the bungee cord.
Total Potential Energy = Initial Potential Energy + Elastic Potential Energy

The elastic potential energy stored in a spring is given by the formula:
Elastic Potential Energy = 0.5 * stiffness constant * (extension)^2

Since the bungee cord is initially stretched by 10.0 m from its relaxed length of 10.0 m, the extension of the cord is 10.0 m.
Elastic Potential Energy = 0.5 * 80 N/m * (10.0 m)^2
= 0.5 * 80 N/m * 100.0 m^2
= 4000 J

Total Potential Energy = Initial Potential Energy + Elastic Potential Energy
= 9800 J + 4000 J
= 13800 J

Step 3: Calculate the maximum distance below the bridge:
The maximum distance below the bridge can be found by equating the total potential energy to the final potential energy at maximum displacement below the bridge.
Potential Energy = mass * gravity * distance

Therefore, 13800 J = 100 kg * 9.8 m/s^2 * distance

Simplifying the equation:
13800 J = 980 N * distance

Finally, solving for the distance:
distance = 13800 J / 980 N
= 14.08 m

Therefore, the man will reach a maximum distance of approximately 14.08 meters below the bridge during his bungee jump.