A 75kg person jumps off a tower with a bungee cord tied around their leg. The cord obeys Hooke's Law and has a cord constant value of k=50 Nts/m. After falling a distance of 15m, the cord begins to stretch. Neglecting air resistance and the mass of the cord,

a) calculate how far below the tower the person falls when coming to a complete stop;
b) calculate the maximum acceleration felt by the person.

We are having trouble since there are two unknowns, how much the cord stretches, and the total distance the person fell.

To solve this problem, we need to use the concept of conservation of energy. The total mechanical energy of the person-bungee system remains constant throughout the fall. At the highest point, all the energy is potential energy, and at the lowest point, all the energy is converted to potential energy.

First, let's find the initial potential energy of the person when they jump off the tower. The formula for potential energy is given by:

PE_initial = m * g * h

where m is the mass of the person (75 kg), g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the initial height from which the person jumps (the distance fallen before the bungee cord starts to stretch).

PE_initial = 75 kg * 9.8 m/s^2 * h_initial

Next, let's find the potential energy of the person when they come to a complete stop. Since the cord is stretched, we need to take into account the potential energy stored in the cord.

PE_final = m * g * h_final + 0.5 * k * x^2

where h_final is the distance below the tower when the person comes to a stop (this is what we need to find), and x is the amount by which the cord stretches.

The total mechanical energy, E, remains constant:

E = PE_initial = PE_final

Substituting the expressions for PE_initial and PE_final:

75 kg * 9.8 m/s^2 * h_initial = 75 kg * 9.8 m/s^2 * h_final + 0.5 * k * x^2

Simplifying and rearranging the equation:

h_initial = h_final + 0.5 * (k / (75 kg * 9.8 m/s^2)) * x^2

This equation relates the initial height to the final height and the amount by which the cord stretches. Since we know the values of h_initial (15 m) and k (50 N/m), we can solve this equation to find h_final.

To find the maximum acceleration felt by the person, we can use Newton's second law. At any given point, the net force acting on the person is equal to their mass times their acceleration.

At the lowest point, when the person comes to a complete stop, the net force can be calculated as:

F_net = m * g + k * x

Since the person comes to a stop when the net force is zero, we can set F_net to zero and solve for x. Then we can calculate the maximum acceleration using a = F_net / m.

By solving these equations simultaneously, we can find the values of h_final and the maximum acceleration felt by the person.