A 63.6-kg bungee jumper is standing on a tall platform (h0 = 50.6 m). The bungee cord has an unstrained length of L0 = 9.54 m and, when stretched, behaves like an ideal spring with a spring constant of k = 67.8 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.

potential energy lost in fall = potential energy gained by our perfect cord

m g (fall distance) = (1/2) k (stretch distance)^2

63.6(9.81)(fall) = (1/2)(67.8)(fall-9.54)^2

623.9 f = 33.9 (f-9.54)^2

18.4 f = f^2 - 19.1 f + 91.0

f^2 - 19.1 f + 72.6 = 0

f = [ 19.1 +/- sqrt (365 - 290) ]/2

f = (19.1 +/- sqrt 75 )/2

f = (19.1 +/- 8.7) /2
= 13.9 or 5.2
5.2 is no good, shorter than cord unstretched
so
f = 13.9
50.6 - 13.9 = 36.7 above water

check my arithmetic !!!

Thank You So Much!

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

1. First, let's find the potential energy of the bungee jumper at the initial position:

PE_initial = m * g * h0

where
m = mass of the bungee jumper = 63.6 kg
g = acceleration due to gravity = 9.8 m/s²
h0 = initial height = 50.6 m

Substituting the values:

PE_initial = 63.6 kg * 9.8 m/s² * 50.6 m
= 31004.32 kg·m²/s²

2. Next, let's find the potential energy at the lowest point of the fall.

At the lowest point, all the potential energy is converted into elastic potential energy of the bungee cord:

PE_elastic = (1/2) * k * (L - L0)²

where
k = spring constant of the bungee cord = 67.8 N/m
L0 = initial length of the bungee cord = 9.54 m
L = length of the bungee cord when the jumper reaches the lowest point

Since all the potential energy is converted into elastic potential energy, we can equate the two:

PE_initial = PE_elastic

3. Solving the equation for L:

31004.32 kg·m²/s² = (1/2) * 67.8 N/m * (L - 9.54 m)²

Rearranging the equation:

(L - 9.54 m)² = (2 * 31004.32 kg·m²/s²) / 67.8 N/m
(L - 9.54 m)² = 912.166279 kg·m²/N
L - 9.54 m = sqrt(912.166279 kg·m²/N)
L - 9.54 m = 30.2 m

Solving for L:

L = 9.54 m + 30.2 m
= 39.74 m

The length of the bungee cord when the jumper reaches the lowest point is approximately 39.74 m.

4. Finally, to find how far the bungee jumper is from the water, we need to subtract the length of the bungee cord from the initial height:

Distance from water = h0 - L
= 50.6 m - 39.74 m
= 10.86 m

Therefore, the bungee jumper is approximately 10.86 meters from the water when he reaches the lowest point in his fall.

To determine the distance of the bungee jumper from the water when he reaches the lowest point in his fall, we need to calculate the amount the bungee cord stretches and subtract it from the initial height.

1. First, let's calculate the gravitational potential energy of the bungee jumper at the starting point. We can use the formula:

Potential Energy = mass * gravity * height

where mass (m) = 63.6 kg, gravity (g) = 9.8 m/s^2, and height (h0) = 50.6 m.

Potential Energy = 63.6 kg * 9.8 m/s^2 * 50.6 m = 31,400.88 J (rounded to four decimal places)

2. Next, we need to calculate the potential energy when the bungee jumper is at the lowest point of his fall. At this point, all the potential energy is converted to elastic potential energy stored in the bungee cord.

Potential Energy = 0.5 * k * (L - L0)^2

where k = spring constant = 67.8 N/m and L0 = unstrained length of the bungee cord = 9.54 m.

L = L0 + (2 * h0) (This equation calculates the total length of the bungee cord when fully stretched)

L = 9.54 m + (2 * 50.6 m) = 110.74 m

Potential Energy = 0.5 * 67.8 N/m * (110.74 m - 9.54 m)^2 = 47,763.48 J (rounded to two decimal places)

3. Now, we can determine the amount the bungee cord stretches by subtracting the potential energy at the lowest point from the initial potential energy:

Stretch = (Potential Energy - Potential Energy at Lowest Point) / k

Stretch = (31,400.88 J - 47,763.48 J) / 67.8 N/m = -241.66 m (negative sign indicates compression)

4. Finally, to find the distance of the bungee jumper from the water at the lowest point, we subtract the magnitude of the stretch from the initial height:

Distance from water = |Stretch| + h0

Distance from water = |-241.66 m| + 50.6 m = 292.26 m

Therefore, the bungee jumper is approximately 292.26 meters from the water when he reaches the lowest point in his fall.