A surfer is catching a wave. Suppose she starts at the top of a wave with a speed of 1.37 m/s and moves down the wave until her speed increases to 9.90 m/s. The drop in her vertical height is 2.64 m. If her mass is 63.3 kg, how much work is done by the (nonconservative) force of the wave?

Work done by the wave equals kinetic energy increase minus gravitational potential energy loss.

W = (M/2) (V2^2 - V1^)2 - M*g*deltaH

where deltaH = 1.27 m

To find the work done by the nonconservative force of the wave, we need to use the work-energy principle.

The work done by the force is equal to the change in the surfer's total mechanical energy.

The total mechanical energy is given by the sum of kinetic energy (KE) and potential energy (PE):

Total mechanical energy = KE + PE

In this case, we'll consider the initial position (at the top of the wave) as the reference level for potential energy.

So, at the top of the wave, the surfer has maximum potential energy and no kinetic energy. At the bottom of the wave, the surfer has maximum kinetic energy and no potential energy.

At the top of the wave:
PE_initial = m * g * h_initial
where m is the mass of the surfer (63.3 kg), g is the acceleration due to gravity (9.8 m/s^2), and h_initial is the initial vertical height (0 m).

At the bottom of the wave:
KE_final = (1/2) * m * v_final^2
where v_final is the final speed of the surfer at the bottom of the wave (9.90 m/s).

The work done by the nonconservative force is given by the change in total mechanical energy:

Work = Total mechanical energy_final - Total mechanical energy_initial
= KE_final + PE_final - KE_initial - PE_initial

Since the initial height is 0, we can simplify the equation:

Work = KE_final + PE_final - PE_initial
= KE_final - PE_initial
= (1/2) * m * v_final^2 - m * g * h_initial

Substituting the given values:
Work = (1/2) * 63.3 kg * (9.90 m/s)^2 - 63.3 kg * 9.8 m/s^2 * 0 m

Simplifying the equation:
Work = (1/2) * 63.3 kg * (9.90 m/s)^2
= 31.65 kg * (9.90 m/s)^2
= 31.65 * 98.01 J
= 3102.76 J

Therefore, the work done by the nonconservative force of the wave is 3102.76 Joules.

To find the work done by the nonconservative force of the wave, we need to calculate the change in the surfer's mechanical energy.

The mechanical energy of the surfer is the sum of her kinetic energy (K) and potential energy (U). According to the work-energy theorem, the change in mechanical energy (ΔE) is equal to the work (W) done on the surfer.

ΔE = W

The change in mechanical energy can be calculated as follows:

ΔE = K2 + U2 - (K1 + U1)

Where K1 is the initial kinetic energy, K2 is the final kinetic energy, U1 is the initial potential energy, and U2 is the final potential energy.

The kinetic energy is given by the equation:

K = (1/2)mv^2

Where m is the mass and v is the velocity.

The potential energy is given by the equation:

U = mgh

Where m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height.

Now, let's calculate the initial and final kinetic energies:

K1 = (1/2) * 63.3 kg * (1.37 m/s)^2
K2 = (1/2) * 63.3 kg * (9.90 m/s)^2

Next, we calculate the initial and final potential energies:

U1 = 63.3 kg * 9.8 m/s^2 * 0 m (since the initial height is 0)
U2 = 63.3 kg * 9.8 m/s^2 * (-2.64 m)

Now, we substitute the values into the equation for the change in mechanical energy:

ΔE = K2 + U2 - (K1 + U1)

Finally, calculate the work done by the nonconservative force of the wave:

W = ΔE

Plug in the values and perform the calculations to find the work done.