Long Slides and Friction

Problem

After having a blast on the slide in Problem 3 and feeling great, A decided to try building an even longer, longer, longer slide.

The slide is a straight line with a height of $h$h above the ground and an angle of inclination of $\theta$θ, and it has no curves or other features.

Since riding it normally would be slow and cause a sore bottom due to friction, A decided to turn it into a roller slide and ride on a piece of cardboard. (This setting is an after-the-fact addition to make $\mu$μ physically plausible, so it’s not really relevant.)

Let $\mu$μ be the coefficient of kinetic friction between the cardboard and the rollers.

A’s mass has decreased again because they were so preoccupied with the confession in Question 2 that they forgot to eat.

*Note: The rollers on this roller slide are assumed not to rotate.

Constraints

$h=50\ \mathrm{m}$h=50 m

$\theta=20^{\circ}$θ=20∘

$\mu=0.08$μ=0.08

$m_0=68\ \mathrm{kg}$m0​=68 kg

$\mathrm{g}=9.8\mathrm{m/s^2}$g=9.8m/s2

$\sin \theta = 0.342$sinθ=0.342

$\cos \theta = 0.94$cosθ=0.94

$\tan \theta = 0.364$tanθ=0.364

$c_w = 4.2\ \mathrm{J/g\cdot K}$cw​=4.2 J/g⋅K

Answer Format

Let $A_0$A0​ be the total frictional heat (energy lost) generated between the cardboard and the rollers while Person A slides down this slide from the top to the bottom.

$\left(\left\lfloor \dfrac{A_0}{100} \right\rfloor\right)\cdot 100$(⌊100A0​​⌋)⋅100

Provide your answer as this expression.