Sure! Let's go through the calculations step by step.
(a) Gravitational Potential Energy (GPE)
The formula for gravitational potential energy is:
\[ \text{GPE} = m \cdot g \cdot h \]
Where:
- \( m = 65 , \text{kg} \) (mass of the diver)
- \( g = 9.8 , \text{N/kg} \) (acceleration due to gravity)
- \( h = 10.0 , \text{m} \) (height above water)
Plugging in the values:
\[ \text{GPE} = 65 , \text{kg} \cdot 9.8 , \text{N/kg} \cdot 10.0 , \text{m} = 6370 , \text{J} \]
(b) Kinetic Energy (KE)
The formula for kinetic energy is:
\[ \text{KE} = \frac{1}{2} m v^2 \]
Where:
- \( m = 65 , \text{kg} \) (mass of the diver)
- \( v = 14 , \text{m/s} \) (speed of the diver at the surface)
Plugging in the values:
\[ \text{KE} = \frac{1}{2} \cdot 65 , \text{kg} \cdot (14 , \text{m/s})^2 = \frac{1}{2} \cdot 65 \cdot 196 = 6370 , \text{J} \]
(c) Comparison of GPE and KE
Both the gravitational potential energy (6370 J) and the kinetic energy (6370 J) are equal in this case. This is consistent with the principle of conservation of energy, which states that energy cannot be created or destroyed in an isolated system. As the diver descends from the height of 10.0 m, the gravitational potential energy is converted into kinetic energy. With no air resistance, all the potential energy at the top is transformed into kinetic energy just before hitting the water.
(d) Speed of the Diver 5.0 m Above the Water
First, we need to calculate the gravitational potential energy at 5.0 m:
\[ \text{GPE}{5.0} = m \cdot g \cdot h{5.0} = 65 \cdot 9.8 \cdot 5.0 = 3175 , \text{J} \]
The total mechanical energy at the top is equal to the mechanical energy at 5.0 m above the water:
\[ \text{Total Energy} = \text{GPE_{top}} = 6370 , \text{J} \]
At 5.0 m above the water, the mechanical energy is the sum of the potential energy and kinetic energy:
\[ \text{Total Energy} = \text{GPE}{5.0} + \text{KE}{5.0} \]
Substituting in the known values:
\[ 6370 , \text{J} = 3175 , \text{J} + \text{KE}_{5.0} \]
Solving for \(\text{KE}_{5.0}\):
\[ \text{KE}_{5.0} = 6370 - 3175 = 3195 , \text{J} \]
Now, using the kinetic energy formula to find the speed at 5.0 m above the water:
\[ \text{KE}{5.0} = \frac{1}{2} m v{5.0}^2 \]
\[ 3195 = \frac{1}{2} \cdot 65 \cdot v_{5.0}^2 \]
Solving for \(v_{5.0}^2\):
\[ v_{5.0}^2 = \frac{3195 \cdot 2}{65} = \frac{6390}{65} \approx 98.38 \]
\[ v_{5.0} \approx \sqrt{98.38} \approx 9.92 , \text{m/s} \]
Summary of Results:
- (a) GPE: 6370 J
- (b) KE: 6370 J
- (c) Comparison: Both values are equal due to conservation of energy.
- (d) Speed 5.0 m above water: approximately 9.92 m/s.
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