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Sagot :
Let's solve the given problem step-by-step to find the height from which a block of ice must be dropped to melt completely, given that 20% of the energy during the fall is retained by the ice. The latent heat of fusion of ice ([tex]\( L \)[/tex]) is 80 cal/g.
### Step 1: Calculate the Energy Required to Melt the Ice
The energy required to completely melt 1 gram of ice is given by its latent heat of fusion.
[tex]\[ \text{Energy required} = L \times \text{mass} \][/tex]
[tex]\[ \text{Energy required} = 80 \text{ cal/g} \times 1 \text{ g} \][/tex]
[tex]\[ \text{Energy required} = 80 \text{ cal} \][/tex]
### Step 2: Convert the Energy Required to Joules
The energy required is initially given in calories. We need to convert this to joules since the potential energy due to gravity will naturally be calculated in joules. The conversion factor is:
[tex]\[ 1 \text{ cal} = 4.186 \text{ joules} \][/tex]
So,
[tex]\[ \text{Energy required in joules} = 80 \text{ cal} \times 4.186 \text{ J/cal} \][/tex]
[tex]\[ \text{Energy required in joules} = 334.88 \text{ J} \][/tex]
### Step 3: Calculate the Energy Retained by the Ice
Given that only 20% of the energy is retained by the ice, we must compute the portion of energy that contributes to melting the ice.
[tex]\[ \text{Energy retained} = \text{20\% of Energy required in joules} \][/tex]
[tex]\[ \text{Energy retained} = 0.20 \times 334.88 \text{ J} \][/tex]
[tex]\[ \text{Energy retained} = 66.976 \text{ J} \][/tex]
### Step 4: Calculate the Height
To find the height from which the ice block must be dropped, we use the principle of conservation of energy. The potential energy (PE) lost by the ice block when it falls from height [tex]\( h \)[/tex] is converted to thermal energy that melts the ice.
[tex]\[ \text{Potential energy (PE) = Mass} \times \text{Gravitational acceleration} \times \text{Height} \][/tex]
[tex]\[ PE = mgh \][/tex]
Where:
- [tex]\( m \)[/tex] is the mass of the ice block (1 g or 0.001 kg),
- [tex]\( g \)[/tex] is the acceleration due to gravity (9.8 m/s²),
- [tex]\( h \)[/tex] is the height we need to find.
Since the energy retained is the potential energy converted,
[tex]\[ \text{Energy retained} = mgh \][/tex]
[tex]\[ 66.976 = 0.001 \times 9.8 \times h \][/tex]
[tex]\[ h = \frac{66.976}{0.001 \times 9.8} \][/tex]
[tex]\[ h = \frac{66.976}{0.0098} \][/tex]
[tex]\[ h \approx 6.834 \text{ meters} \][/tex]
### Final Answer
The block of ice must be dropped from a height of approximately 6.834 meters for it to melt completely, given that 20% of the energy during the fall is retained by the ice.
### Step 1: Calculate the Energy Required to Melt the Ice
The energy required to completely melt 1 gram of ice is given by its latent heat of fusion.
[tex]\[ \text{Energy required} = L \times \text{mass} \][/tex]
[tex]\[ \text{Energy required} = 80 \text{ cal/g} \times 1 \text{ g} \][/tex]
[tex]\[ \text{Energy required} = 80 \text{ cal} \][/tex]
### Step 2: Convert the Energy Required to Joules
The energy required is initially given in calories. We need to convert this to joules since the potential energy due to gravity will naturally be calculated in joules. The conversion factor is:
[tex]\[ 1 \text{ cal} = 4.186 \text{ joules} \][/tex]
So,
[tex]\[ \text{Energy required in joules} = 80 \text{ cal} \times 4.186 \text{ J/cal} \][/tex]
[tex]\[ \text{Energy required in joules} = 334.88 \text{ J} \][/tex]
### Step 3: Calculate the Energy Retained by the Ice
Given that only 20% of the energy is retained by the ice, we must compute the portion of energy that contributes to melting the ice.
[tex]\[ \text{Energy retained} = \text{20\% of Energy required in joules} \][/tex]
[tex]\[ \text{Energy retained} = 0.20 \times 334.88 \text{ J} \][/tex]
[tex]\[ \text{Energy retained} = 66.976 \text{ J} \][/tex]
### Step 4: Calculate the Height
To find the height from which the ice block must be dropped, we use the principle of conservation of energy. The potential energy (PE) lost by the ice block when it falls from height [tex]\( h \)[/tex] is converted to thermal energy that melts the ice.
[tex]\[ \text{Potential energy (PE) = Mass} \times \text{Gravitational acceleration} \times \text{Height} \][/tex]
[tex]\[ PE = mgh \][/tex]
Where:
- [tex]\( m \)[/tex] is the mass of the ice block (1 g or 0.001 kg),
- [tex]\( g \)[/tex] is the acceleration due to gravity (9.8 m/s²),
- [tex]\( h \)[/tex] is the height we need to find.
Since the energy retained is the potential energy converted,
[tex]\[ \text{Energy retained} = mgh \][/tex]
[tex]\[ 66.976 = 0.001 \times 9.8 \times h \][/tex]
[tex]\[ h = \frac{66.976}{0.001 \times 9.8} \][/tex]
[tex]\[ h = \frac{66.976}{0.0098} \][/tex]
[tex]\[ h \approx 6.834 \text{ meters} \][/tex]
### Final Answer
The block of ice must be dropped from a height of approximately 6.834 meters for it to melt completely, given that 20% of the energy during the fall is retained by the ice.
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