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 82 kg, and the height of the water slide is 11.3 m. If the kinetic frictional force does -6.2 × 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 need to apply the principle of conservation of mechanical energy.
The total mechanical energy at the top of the slide is given by the sum of the potential energy and the initial kinetic energy:
E_top = PE_top + KE_top
where PE_top is the potential energy at the top and KE_top is the kinetic energy at the top.
The total mechanical energy at the bottom of the slide is given by:
E_bottom = PE_bottom + KE_bottom
where PE_bottom is the potential energy at the bottom (which is zero) and KE_bottom is the kinetic energy at the bottom.
According to the principle of conservation of mechanical energy, the total mechanical energy should remain constant throughout the slide. Therefore:
E_top = E_bottom
PE_top + KE_top = PE_bottom + KE_bottom
Since the student starts from rest, the initial kinetic energy (KE_top) is zero. Therefore:
PE_top = KE_bottom
The potential energy at the top of the slide can be calculated as:
PE_top = m * g * h
where m is the mass of the student, g is the acceleration due to gravity, and h is the height of the slide.
Substituting the given values:
PE_top = 82 kg * 9.81 m/s^2 * 11.3 m
Next, we need to find the kinetic energy at the bottom of the slide (KE_bottom). The work done by the kinetic frictional force (W_friction) can be expressed as the change in energy:
W_friction = KE_bottom - KE_top
Given that the work done by the frictional force is -6.2 × 10^3 J, we can rearrange the equation:
KE_bottom = W_friction + KE_top
Substituting the given values:
KE_bottom = -6.2 × 10^3 J + 0
Finally, we can calculate the speed of the student at the bottom of the slide using the relationship between kinetic energy and speed:
KE_bottom = ½ * m * v^2
where v is the speed of the student at the bottom of the slide.
Substituting the known values:
-6.2 × 10^3 J = ½ * 82 kg * v^2
Now, you can solve this equation to find the value of v, the speed of the student at the bottom of the slide.