assume that a hailstone at 0 degrees celsius falls through air at a uniform temperature of 0 degrees celsius and lands on a sidewalk also at this temperature. from what initial height must the stone fall in order to entirely melt upon impact?
To determine the initial height at which the hailstone must fall in order to entirely melt upon impact, we need to consider the amount of heat transferred during the process. The heat transferred can be calculated using the specific heat capacity of ice and the latent heat of fusion for ice.
Here's how you can approach the problem step-by-step:
1. Find the amount of heat required to raise the temperature of the hailstone from -1°C (slightly below freezing) to 0°C.
- The specific heat capacity of ice is 2.09 J/g°C.
- Determine the mass of the hailstone. Let's assume it is 100 grams, for example.
- Calculate the heat required using the formula: Q = m * c * ΔT, where Q is the heat transferred, m is the mass, c is the specific heat capacity, and ΔT is the change in temperature.
- Substituting the values, Q = 100 g * 2.09 J/g°C * (0 - (-1)) °C.
2. Determine the amount of heat required for the phase change from solid (ice) to liquid (water).
- The latent heat of fusion for ice is 334 J/g.
- Calculate the heat required using the formula: Q = m * L, where Q is the heat transferred, m is the mass, and L is the latent heat of fusion.
- Substituting the values, Q = 100 g * 334 J/g.
3. Add the heat required for the temperature change and the phase change together to find the total heat transfer.
- Total heat transfer = heat required for temperature change + heat required for phase change.
4. Finally, equate this heat transfer to the potential energy (mgh) at the initial height, where m is the mass of the hailstone, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the initial height.
By setting up and solving the equation, you can determine the initial height from which the hailstone must fall in order to entirely melt upon impact.