Question

(I) A 16.0-kg child descends a slide 2.20 m high and, starting from rest, reaches the bottom with a speed of 1.25 m/s. How much thermal energy due to friction was generated in this process?

Short answer

The generated thermal energy in this process is 332.46 J.

Step-by-step solution

Given data

The generated thermal energy in a system is equal to the loss of mechanical energy from the system.

Given data:

The mass of the child is $m=16.0\mathrm{k}\mathrm{g}$.

The height of the slide is $h=2.20\mathrm{m}$.

The initial speed of the child at the height is $v_o=0$ as it starts from rest.

The speed of the child at the bottom is $v=1.25\frac{\mathrm{m}}{\mathrm{s}}$.

Let $\mathrm{\Delta }H$ be the generated thermal energy in this process.

Calculation of the thermal energy

The total mechanical energy of the system is calculated as follows:

$\begin{array}{rl}{E}_1& =mgh+\frac{1}{2}m{v}_o^2\\ & =mgh+\left(\frac{1}{2}m\times 0^2\right)\\ & =mgh\end{array}$

The total mechanical energy of the system at the bottom is calculated as follows:

$\begin{array}{rl}{E}_2& =\left(mg\times 0\right)+\frac{1}{2}m{v}^2\\ & =\frac{1}{2}m{v}^2\end{array}$

Now the thermal energy is equal to the loss in mechanical energy. Then,

Now the thermal energy is equal to the loss in mechanical energy. Then,

$\begin{array}{rl}\mathrm{\Delta }H& =E_1-E_2\\ & =mgh-\frac{1}{2}m{v}^2\\ & =\left[\left(16.0\mathrm{k}\mathrm{g}\right)\times \left(9.80\frac{\mathrm{m}}{{\mathrm{s}}^2}\right)\times 2.20\mathrm{m}\right]-\left[\frac{1}{2}\times \left(16.0\mathrm{k}\mathrm{g}\right)\times {\left(1.25\frac{\mathrm{m}}{\mathrm{s}}\right)}^2\right]\\ & =332.46\mathrm{J}\end{array}$

Hence, the generated thermal energy in this process is 332.46 J.