A block of ice at 0^\circ {\rm C} is dropped from a height that causes it to completely melt upon impact. Assume that there is no air resistance and that all the energy goes into melting the ice. What is the height necessary for this to occur.
Q=λm
E=mgh
Q=E
λm= mgh
h=λ/g = 335000/9.8 = 34184 m
To determine the height necessary for a block of ice to completely melt upon impact, we need to consider the concept of potential energy, kinetic energy, and energy conservation.
When the block of ice is dropped, it initially has gravitational potential energy due to its height. As it falls, this potential energy is converted into kinetic energy. Upon impact, all the kinetic energy is used to melt the ice. We need to find the height where the potential energy is equal to the energy required to melt the ice.
The energy required to completely melt the ice can be calculated using the specific heat capacity and the latent heat of fusion of ice.
1. Calculate the energy required to melt the ice:
- The specific heat capacity of ice is approximately 2.09 J/g°C.
- The latent heat of fusion of ice is approximately 334 J/g.
2. Determine the mass of the ice block.
- Assume a mass of 'm' grams for the ice block.
3. Determine the change in temperature.
- The initial temperature is 0°C, and the final temperature after melting is 0°C.
- Therefore, the change in temperature, ΔT, is 0 - 0 = 0°C.
4. Calculate the energy required to change the temperature from 0°C to 0°C.
- The formula for energy is Q = mcΔT, where 'Q' is the energy, 'm' is the mass, 'c' is the specific heat capacity, and 'ΔT' is the change in temperature.
- Since ΔT = 0, the energy required for the temperature change is 0.
5. Calculate the energy required for melting the ice.
- The formula for energy is Q = mL, where 'Q' is the energy, 'm' is the mass, and 'L' is the latent heat of fusion.
- Therefore, the energy required for melting = m * 334 J/g.
6. Equate the potential energy to the energy required for melting.
- At height 'h', the potential energy is given by PE = mgh, where 'm' is the mass, 'g' is the acceleration due to gravity (9.8 m/s²), and 'h' is the height.
- Set PE = energy required for melting, i.e., mgh = m * 334 J/g.
7. Cancel out the mass, and solve for the height 'h'.
- h = 334 J/g / (9.8 m/s²)
By performing the calculations, we can obtain the height necessary for the block of ice to completely melt upon impact.