A student, starting from rest, slides down a water slide. On the way down, a kinetic frictional force (a nonconservative force) acts on her. The student has a mass of 73 kg, and the height of the water slide is 12.0 m. If the kinetic frictional force does -6.9 × 103 J of work, how fast is the student going at the bottom of the slide? Use g = 9.81 m/s2

To find the speed of the student at the bottom of the slide, we can use the principle of conservation of energy. We need to consider the initial potential energy of the student at the top of the slide, the work done by the frictional force, and the final kinetic energy of the student at the bottom of the slide.

1. Calculate the initial potential energy:
The initial potential energy (PEi) is given by the equation PEi = mgh, where m is the mass of the student, g is the acceleration due to gravity, and h is the height of the slide.
PEi = (73 kg)(9.81 m/s^2)(12.0 m)

2. Calculate the work done by the frictional force:
The work done by the frictional force (Wf) is given as negative because it opposes the motion.
Wf = -6.9 × 10^3 J

3. Calculate the final kinetic energy:
The final kinetic energy (KEf) is given by the equation KEf = 0.5mv^2, where v is the final velocity.
KEf = 0.5(73 kg)v^2

4. Apply the conservation of energy principle:
According to the principle of conservation of energy, the initial potential energy plus the work done is equal to the final kinetic energy.
PEi + Wf = KEf

Substituting the values we have:
(73 kg)(9.81 m/s^2)(12.0 m) - 6.9 × 10^3 J = 0.5(73 kg)v^2

5. Solve for v:
Rearrange the equation to solve for v:
0.5(73 kg)v^2 = (73 kg)(9.81 m/s^2)(12.0 m) - 6.9 × 10^3 J

Divide both sides by 0.5(73 kg) to isolate v^2:
v^2 = ((73 kg)(9.81 m/s^2)(12.0 m) - 6.9 × 10^3 J) / (0.5(73 kg))

Take the square root of both sides to solve for v:
v = √[((73 kg)(9.81 m/s^2)(12.0 m) - 6.9 × 10^3 J) / (0.5(73 kg))]

By plugging in the given values and evaluating the expression, we can find the speed (v) at the bottom of the slide.