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 78 kg, and the height of the water slide is 12.5 m. If the kinetic frictional force does -5.4 × 103 J of work, how fast is the student going at the bottom of the slide?
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To find the speed of the student at the bottom of the slide, we can use the principle of conservation of mechanical energy. The total mechanical energy of the student at the top of the slide is equal to the sum of the potential energy and the initial kinetic energy:
Initial mechanical energy = Potential energy + Initial kinetic energy
The potential energy of the student at the top of the slide is given by:
Potential energy = mass × gravity × height
where mass is the mass of the student (78 kg), gravity is the acceleration due to gravity (9.8 m/s²), and height is the height of the water slide (12.5 m).
Next, we need to determine the work done by the kinetic frictional force. Since work is the transfer of energy from one form to another, the work done by the frictional force is equal to the change in mechanical energy of the student:
Work done by frictional force = Final mechanical energy - Initial mechanical energy
In this case, the work done by the frictional force is given as -5.4 × 10³ J, indicating that it takes energy away from the system.
Now, we can rearrange the equation to solve for the final mechanical energy:
Final mechanical energy = Initial mechanical energy + Work done by frictional force
Finally, we equate the final mechanical energy to the sum of the potential energy and the final kinetic energy to obtain the speed of the student at the bottom of the slide:
Potential energy + Final kinetic energy = Final mechanical energy
The final kinetic energy is given by:
Final kinetic energy = (1/2) × mass × velocity²
where mass is the mass of the student (78 kg) and velocity is the speed of the student at the bottom of the slide (which we want to find).
Now, we can plug in the given values and solve for velocity.