Question

A river descends 15 mthrough rapids. The speed of the water is 3.2 m/supon entering the rapids and 13 m/supon leaving. What percentage of the gravitational potential energy of the water–Earth system is transferred to kinetic energy during the descent? (Hint: Consider the descent of, say, 10 kgof water.)

Short answer

The percentage of the gravitational potential energy of water transferred to kinetic energy during the descent is 54% .

Step-by-step solution

The given data

i) The height at which the river descends, h = 15m

ii) Initial velocity of the river,$v_i=3.2m/s$

iii) Final velocity of the river,$v_f=13m/s$

iv) Mass of water,$m=10kg$

Understanding the concept of energy 

We use the concept of kinetic energy and gravitational potential energy. We find the gravitational potential energy and kinetic energy, and then we can take its ratio to find the percentage.

Formulae:

The potential energy at a height, PE = mgh (i)

The kinetic energy of the body, $KE=\frac{1}{2}m{v}^2$ (ii)

Calculation of percentage of gravitational potential energy transferred to the kinetic energy

Gravitational potential energy of the water is given using equation (i) as follows:

$$\begin{gathered} P.E=10\times 9.8\times 15 \\ =1470J \\ =1.47\times {10}^{3}J \end{gathered}$$

Change in kinetic energy is given using equation (ii) as follows:

$$\begin{gathered} K.E=\frac{1}{2}m{v}_f^2-\frac{1}{2}m{v}_i^2 \\ =\frac{1}{2}\left(10\right){\left(13\right)}^2-\frac{1}{2}\left(10\right){\left(3.2\right)}^2 \\ =0.79\times {10}^{3}J \end{gathered}$$

So, the percentage of gravitational potential energy transferred to the kinetic energy is given as:

$$\begin{gathered} percentage=\frac{K.E}{P.E}\times 100 \\ =\frac{0.79\times {10}^3}{1.47\times {10}^3}\times 100 \\ =54\% \end{gathered}$$

Hence, the value of the required percentage is 54% .