A 61.0 -kg bungee jumper is standing on a tall platform (h0 = 46.0 m), as indicated in the figure. The bungee cord has an unstrained length of L0 = 9.00 m, and when stretched, behaves like an ideal spring with a spring constant of 76.0 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.
1/2mghi+1/2kx^2i=1/2mghf+1/2kx^2f
1/2(mg-hf)+1/2kx^2i/1/2k=x^2f
1/2(61)(9.80)
im stuck
He has no kinetic energy when he starts the jump and none when he gets to the lowest point.
therefore all potential energy lost m g (Ho-h) goes into (1/2) k x^2
x = 46 - 9 - h = 37 - h
m g (46-h) = (1/2)k(37-h)^2
61 * 9.8 (46-h) = 38 (37-h)^2
To solve for the distance the bungee jumper is from the water when he reaches the lowest point, we can use the principle of conservation of mechanical energy.
The mechanical energy is conserved in this system, which means that the sum of the potential energy and the kinetic energy at the initial point (h0) is equal to the sum of the potential energy and the kinetic energy at the final point (hf).
We can start by calculating the initial potential energy (U_initial) at height h0. The formula for potential energy is U = m * g * h, where m is the mass of the bungee jumper, g is the acceleration due to gravity (9.8 m/s^2), and h is the height.
U_initial = m * g * h0
= (61.0 kg) * (9.8 m/s^2) * (46.0 m)
Next, we calculate the initial potential energy stored in the spring (Us_initial) at the extended length L0. The formula for the potential energy stored in an ideal spring is Us = (1/2) * k * x^2, where k is the spring constant and x is the displacement from the equilibrium position.
Us_initial = (1/2) * k * L0^2
= (1/2) * (76.0 N/m) * (9.00 m)^2
Now, we can solve for the final potential energy (U_final) and the final potential energy stored in the spring (Us_final) at the lowest point.
Since the bungee cord stretches, the final length (Lf) can be found by subtracting the compressed length of the spring from the initial length: Lf = L0 + x.
Using the conservation of mechanical energy, the equation becomes:
U_initial + Us_initial = U_final + Us_final
Plugging in the initial and final values:
(m * g * h0) + (1/2 * k * L0^2) = (m * g * hf) + (1/2 * k * Lf^2)
From the information given in the problem, we know the mass (m), g, h0, L0, and k. We need to solve for hf, so we rearrange the equation and isolate hf on one side:
(m * g * h0) + (1/2 * k * L0^2) - (1/2 * k * Lf^2) = m * g * hf
Now, substitute the known values into the equation and solve for hf.