Final answer:
The student's speed at the bottom of the water slide can be calculated using the conservation of energy principle and the work done by friction. By setting up the energy equation and inserting the known values, we can solve for the velocity at the bottom of the slide.
Step-by-step explanation:
To find how fast the student is going at the bottom of the water slide, we can use the conservation of energy principle taking into account the work done by nonconservative forces such as friction. The total mechanical energy (potential energy + kinetic energy) at the top of the slide will equal the total mechanical energy at the bottom, minus the work done by friction.
At the top of the slide, the student has potential energy (PE) due to their height above the bottom. This can be calculated using the formula PE = m × g × h where m is the mass, g is the acceleration due to gravity (9.8 m/s2), and h is the height. The kinetic energy (KE) at the top is zero because the student starts from rest. At the bottom, all the potential energy will have transferred to kinetic energy minus the work done by friction. The kinetic energy can be expressed as KE = 1/2 × m × v2 where v is the velocity at the bottom of the slide.
The work done by friction is given as a negative value because it removes energy from the system. Therefore, we can set up the energy conservation equation like this:
PEtop + KEtop + Workfriction = KEbottom + PEbottom
Since the potential energy at the bottom is zero (PEbottom = 0) and the kinetic energy at the top is zero (KEtop = 0), the equation simplifies to:
m × g × h + Workfriction = 1/2 × m × v2
Solving this equation for v will give us the velocity at the bottom of the slide.
Using the given values, m = 71 kg, h = 12.5 m, and Workfriction = -7.1×103 J, and substituting g = 9.8 m/s2, we get:
71 kg × 9.8 m/s2 × 12.5 m - 7,100 J = 1/2 × 71 kg × v2
Now we solve for v to find the student's speed at the bottom of the slide.