# Kate's Bungee Jump: Calculating Distance Below Bridge

• rsala
In summary, Kate wants to jump off the edge of a bridge that spans a river below. The bungee cord, which has length L when unstretched, will first straighten and then stretch as Kate falls. The spring constant k is unknown, but the acceleration due to gravity is g. When Kate stops oscillating and comes finally to rest, she will be hanging below the bridge. The energy approach is used to find the value of d where she is closest to hitting the water.
rsala

## Homework Statement

Kate, a bungee jumper, wants to jump off the edge of a bridge that spans a river below. Kate has a mass m, and the surface of the bridge is a height h above the water. The bungee cord, which has length L when unstretched, will first straighten and then stretch as Kate falls.

Assume the following:

The bungee cord behaves as an ideal spring once it begins to stretch, with spring constant k.
Kate doesn't actually jump but simply steps off the edge of the bridge and falls straight downward.
Kate's height is negligible compared to the length of the bungee cord. Hence, she can be treated as a point particle.

Use g for the magnitude of the acceleration due to gravity.

How far below the bridge will Kate eventually be hanging, once she stops oscillating and comes finally to rest? Assume that she doesn't touch the water.

## Homework Equations

conservation of energy

## The Attempt at a Solution

the problem states to use only terms introduced in the problem, so don't use X in kx

$$mgd = \frac{1}{2}k(d-L)^{2}$$

$$\frac{2mgd}{k} = (d-L)^{2}$$

$$\sqrt{\frac{2mgd}{k}} = d - L$$

$$\sqrt{\frac{2mgd}{k}} + L = d$$

$$\sqrt{d} * \sqrt{\frac{2mg}{k}} + L = d$$

$$\sqrt{\frac{2mg}{k}} + \frac{L}{\sqrt{d}} = \frac{d}{\sqrt{d}} = \sqrt{d}$$

$$(\sqrt{\frac{2mg}{k}} + \frac{L}{\sqrt{d}})^{2} = d$$

im stuck here since i have to solve for d, and i don't even think the result will be similar to the answer

the answer is $$\frac{mg}{k} + L$$

where did i go wrong?

Last edited:
Why do you think those energies should be equal? When she is at rest her weight mg is equal to the force of the spring pulling up, k*(d-L). The energy approach is used if you want to find the value of d where she is closest to hitting the water.

Last edited:
wow that was simple, hm

well the next part of the problem asks to find the k constant if she jumps and JUST reachs the water,, is the energy approach, then correct?

BTW how did you know to set the forces equal to each other and not the energys?

thanks

Last edited:
Because of the phrase "once she stops oscillating and comes finally to rest". That implies that frictional forces have been acting on her and dissipating energy. The energy calc wouldn't account for that lost energy. On the other hand, for the 'just reaches the water' case, it's reasonable to assume that there hasn't been that much energy lost to friction, and since KE is zero you can equate the PE's. Now you tell me why I can't just balance the forces in that case?

Last edited:

## 1. How does the height of the bridge affect the distance Kate falls during the bungee jump?

The height of the bridge is a crucial factor in calculating the distance Kate falls during the bungee jump. The higher the bridge, the longer the distance she will fall before the bungee cord reaches its maximum tension and starts to slow her descent.

## 2. What is the formula used to calculate the distance below the bridge?

The formula used to calculate the distance below the bridge is d = 1/2 * g * t^2, where d is the distance, g is the acceleration due to gravity (9.8 m/s^2), and t is the time in seconds. This formula is based on the laws of motion and takes into account the initial velocity, acceleration, and time.

## 3. How does the weight of the bungee cord affect the distance Kate falls?

The weight of the bungee cord does not significantly affect the distance Kate falls during the bungee jump. The main factors that affect the distance are the height of the bridge, the length and elasticity of the bungee cord, and the weight and height of the jumper.

## 4. Is it possible for Kate to reach the water below the bridge during the bungee jump?

No, it is highly unlikely for Kate to reach the water below the bridge during the bungee jump. The bungee cord is designed to stretch and slow down the jumper's descent before reaching the water. Additionally, the length of the bungee cord is calculated to ensure that the jumper does not hit the water.

## 5. How does air resistance affect the distance Kate falls during the bungee jump?

Air resistance plays a minimal role in the distance Kate falls during the bungee jump. The initial velocity of the jumper is usually not high enough for air resistance to have a significant impact. However, it may slightly slow down the descent, resulting in a slightly shorter distance below the bridge.

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