The hero pursues the villain up to the top of a bungee jumping apparatus. The villain appears trapped but to create a diversion she drops a bottle. The hero quickly straps on the bungee cord to his leg and dives straight down to grab the bottle while its still in the air. Determine the length of the unstretched bungee cord. The hero is supposed to grab the bottle before the bungee begins to stretch so that the stretching of the bungee cord will stop him gently. You estimate that the hero can jump off the bungee tower with a maximum velocity of 10 ft/s straight down by pushing off with his feet and can react to the villains dropping the bottle by strapping on the bungee cord and jumping in 2 seconds.

Find the length of the un-stretched bungee cord.

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To determine the length of the unstretched bungee cord, we can use the basic principles of physics and motion. Let's break down the problem step by step:

1. First, we need to find the height of the tower. Since the hero jumps off the tower with a maximum velocity of 10 ft/s and takes 2 seconds to react, we can use the equation of motion:
h = v * t + (1/2) * g * t², where v is the initial velocity (10 ft/s), t is the time (2 seconds), and g is the acceleration due to gravity (-32 ft/s²).

Plugging in the values, we get:
h = 10 ft/s * 2 s + (1/2) * (-32 ft/s²) * (2 s)²
h = 20 ft - 64 ft
h = -44 ft

The negative sign indicates that the height is below the starting point (the top of the tower).

2. Next, we need to consider the free fall of the bottle. The time it takes for the bottle to reach the ground is the same as the time it takes for the hero to jump off the tower and react (2 seconds). The height of the bottle is irrelevant in this case, as the hero needs to grab it before it falls too far to reach.

3. Now, let's consider the bungee cord's function. The hero must grab the bottle before the bungee cord starts to stretch. This means that when the hero reaches the fully stretched position of the bungee cord, he will still be able to grab the bottle. Since the time it takes for the bungee cord to stretch is very short compared to the entire time of the fall, we can neglect it for this calculation.

4. Finally, to find the length of the unstretched bungee cord, we need to consider the total distance covered by the hero. This includes the distance fallen from the tower (44 ft) and the additional distance traveled by the stretched bungee cord.

The length of the bungee cord (unstretched) equals the total distance covered by the hero before the bungee cord starts to stretch. To calculate this, we need to consider the distance traveled during free fall and the extra distance traveled by the fully stretched bungee cord.

Using the equations of motion:
Distance fallen = (1/2) * g * t²
Distance traveled by stretched bungee cord = v * t + (1/2) * a * t²

Plugging in the values:
Distance fallen = (1/2) * (-32 ft/s²) * (2 s)²
Distance fallen = (1/2) * (-32 ft/s²) * 4 s²
Distance fallen = -64 ft

Distance traveled by stretched bungee cord = 10 ft/s * 2 s + (1/2) * 0 * 2 s² (since the bungee cord is fully stretched in this scenario, its acceleration is zero)
Distance traveled by stretched bungee cord = 20 ft

Length of unstretched bungee cord = Distance fallen + Distance traveled by stretched bungee cord
Length of unstretched bungee cord = -64 ft + 20 ft
Length of unstretched bungee cord = -44 ft

Therefore, the length of the unstretched bungee cord is 44 ft.