Question

Two people respectively 100kg and 70kg, are standing on a bridge 100m above water. They are attached to an bungee cable with coefficient of k=3 kN/m. Find Initial length of the cable if its max length is when the people touch the water. Height of people is negligible. Found L initial to be 10.54m. Now I must find the height person 2 (70kg) will reach on the way back up if person one (100kg) were to let go when max length is reached.

R3 70tO 00ka (70) (io0ky)

Answer 1

Taking the water level as the datum point i.e., potential energy is zero at that point.

Initially, (at position 1) both the persons are at 100 m above water, possessing only Potential energy (PE).

Let,

m_a = mass of person A = 70 kg

m_b = mass of person B = 100 kg

PE₁ = m_agH + m_bgH

where,

g = acceleration due to gravity = 9.81 m/s²

H = height of the bridge from water level = 100 m

PE₁ = (70 x 9.81 x 100) + (100 x 9.81 x 100)

PE₁ = 68670 + 98100 = 166770 J

PE₁ = 166770 J or 166.77 kJ

When both the persons reach the water level, it is said that the maximum length of the rope is reached (maximum elongation), means velocity is zero at that point.

That means total energy at point 2 is also Potential energy.

PE₂ = m_agH + m_bgH + kx²/2

H = 0 (at water level)

k = 3 kN/m (given)

x = elongation in cable.

PE₂ = 0 + 0 + 3x²/2

PE₂ = 3x²/2 kJ

As energy is conserved, PE₁ = PE₂

(3/2)x² = 166.77

=> x² = (2/3)(166.77)

=> x = 10.54 m

The elongation in cable is 10.54 m.

The elongated length of cable is 100 m. (as elongated cable just touches the water)

So, initial length of cable, (say L) = 100 - 10.54 = 89.46 m

Initial length of cable, L = 89.46 m

_____________________________

Case - I The person is tied to the cable and can not leave the cable.

Given, on reaching the water surface the person weighing 100 kg has left the rope.

If 100 kg person have not left the rope both will again reach the height of the bridge, as energy is not lost in any form.

Let us break the question for better understanding the problem. (refer figure)

The energy of the both the men initially was m_agH + m_bgH = (70 x 9.81 x 100) + (100 x 9.81 x 100)

All this energy is stored in the cable at position 2.

Now, the 100 kg person has left the cable.

Now, the complete energy (m_agH + m_bgH) is transferred to the person weighing 70 kg.

m_agH (= 70 x 9.81 x 100) amount of energy is required from the energy (m_agH + m_bgH), to take the person to bridge.

Now, on reaching the bridge, we are left with m_bgH (= 100 x 9.81 x 100) amount of energy. This energy is stored in the form of Kinetic energy of the 70 kg person.

KE = 100 x 9.81 x 100 = 98100 J

To reach at a height equal to natural length of the rope (above the bridge), energy required is ;

(70 x 9.81 x 89.46) = 61432.18 J

On reaching a height 89.46 m from the bridge, energy left = 98100 - 61432.18 = 36667.82 J

Now, the rope will start to elongate and lets say it extends by an amount 'y'.

The remaining energy will be converted to Potential energy of the cable and Potential energy of m_a.

=> 36667.82 = (3000/2)y² + (70 x 9.81 x y)

=> 1500y² + 686.7y -36667.82 = 0

=> y = 5.22 m

Therefore, the 70 kg person will reach a height of (89.46 + 5.22) m = 94.68 m from the bridge.

or, 94.68 + 100 = 194.68 m from the water level.

h = 194.68 m from the water level.

________________________________

Case-II If the person let go off the cable on reaching the bridge.

The 100 kg person leaves the rope and as the energy required to pull the 100 kg person is also stored in the cable, the 70 kg person will get that energy.

Let's say the person reach a maximum height 'h' (in meters) from water surface (point 3).

(at that point velocity is zero, potential energy stored in cable is also zero (as the person has left the cable))

The only energy will be potential energy due to elevation 'h'.

PE₃ = m_agh

PE₃ = 70 x 9.81 x h = 686.7h J

As energy is conserved; all the potential energy stored in the cable is transferred to the person.

PE₃ = PE₂ (= PE₁)

686.7h (Joule)= 166770 (Joule)

=> h = 242.86 m

Means the person will reach 242.86 m above from the water surface or (242.86 - 100) m above from the bridge.