A river descends 15 m through rapids. The speed of the water is 3.2 m/s upon entering the rapids and 13 m/s upon 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 kg of water.)

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the change in gravitational potential energy (ΔU)
The formula for gravitational potential energy is given by:

\[ \Delta U = mgh \]

Where:
- \( m \) = mass of the water (we will take \( m = 10 \, \text{kg} \))
- \( g \) = acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \))
- \( h \) = height (which is \( 15 \, \text{m} \))

Substituting the values:

\[ \Delta U = 10 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times 15 \, \text{m} = 1470 \, \text{J} \]

### Step 2: Calculate the change in kinetic energy (ΔK)
The change in kinetic energy can be calculated using the formula:

\[ \Delta K = \frac{1}{2} m v_f^2 - \frac{1}{2} m v_i^2 \]

Where:
- \( v_f \) = final velocity (13 m/s)
- \( v_i \) = initial velocity (3.2 m/s)

Substituting the values:

\[ \Delta K = \frac{1}{2} \times 10 \, \text{kg} \times (13^2 - 3.2^2) \]

Calculating \( 13^2 \) and \( 3.2^2 \):

\[ 13^2 = 169 \quad \text{and} \quad 3.2^2 = 10.24 \]

Now substituting these values back into the equation:

\[ \Delta K = \frac{1}{2} \times 10 \times (169 - 10.24) = \frac{1}{2} \times 10 \times 158.76 = 793.8 \, \text{J} \]

### Step 3: Calculate the percentage of gravitational potential energy converted to kinetic energy
To find the percentage of gravitational potential energy that is converted to kinetic energy, we use the formula:

\[ \text{Percentage} = \left( \frac{\Delta K}{\Delta U} \right) \times 100 \]

Substituting the values we calculated:

\[ \text{Percentage} = \left( \frac{793.8}{1470} \right) \times 100 \]

Calculating this gives:

\[ \text{Percentage} \approx 54% \]

### Final Answer
The percentage of the gravitational potential energy of the water-Earth system that is transferred to kinetic energy during the descent is approximately **54%**.

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