A 50 kg student, starting from rest, slides down an 10.8 meter high water slide. How fast is he going at the bottom of the slide?

To determine the speed of the student at the bottom of the water slide, we can use the principle of conservation of energy. The total mechanical energy at the top of the slide can be equated to the total mechanical energy at the bottom.

Let's break down the problem step by step:

Step 1: Determine the potential energy at the top.
The potential energy (PE) of an object is given by the formula: PE = m * g * h, where m represents the mass (50 kg) of the student, g is the acceleration due to gravity (9.8 m/s²), and h is the height (10.8 m) of the slide.

Substituting the given values into the formula:
PE_top = (50 kg) * (9.8 m/s²) * (10.8 m)

Step 2: Determine the kinetic energy at the bottom.
At the bottom of the slide, all the potential energy is converted into kinetic energy (KE). The kinetic energy of an object is given by the formula: KE = (1/2) * m * v², where v represents the velocity (speed) of the student.

Substituting the given values into the formula:
KE_bottom = (1/2) * (50 kg) * v²

Step 3: Equate potential energy and kinetic energy.
According to the principle of conservation of energy, the potential energy at the top is equal to the kinetic energy at the bottom, i.e.,
PE_top = KE_bottom

Thus, we can set up an equation:
(50 kg) * (9.8 m/s²) * (10.8 m) = (1/2) * (50 kg) * v²

Step 4: Solve for the velocity.
Simplifying the equation:
(50 kg) * (9.8 m/s²) * (10.8 m) = (1/2) * (50 kg) * v²
Multiply and cancel the units:
5292 N·m = 25 kg·v²

Divide both sides by 25 kg:
211.68 N·m/kg = v²

Take the square root of both sides to find v:
v = √211.68 N·m/kg

Calculating this, the final answer is approximately:
v ≈ 14.55 m/s

So, the student is going at around 14.55 m/s at the bottom of the slide.