An 8.56 kg block of ice at 0 degrees celsius is sliding on a rough horizontal icehouse floor (also at 0 degrees celsius) at 15.6 m/s. Assume that half of any heat generated goes into the floor and the rest goes into the ice. How much ice (in kg) has melted after the speed of the ice has been reduced to 10.2 m/s? What is the maximum amount of ice that will melt?
Use energy considerations.
m=8.56 kg
v0=15.6 m/s,
v1=10.2 m/s
latent heat of fusion of ice
= 333.55 KJ/kg
Initial kinetic energy
=(1/2)mv0²
Final kinetic energy
=(1/2)mv1²
Half of energy lost goes to melt the ice block
Ef=(1/2)m(v0²-v1²)
=(1/2)8.56(15.6²-10.2²)
= 596.3 J
Amount of melted ice
= ((596.3/2)/1000)KJ /333.55 (KJ/kg)
= 0.9 g
Maximum amount of ice would melt if all the kinetic loss goes to melting the ice.
To solve this problem, we need to understand the concept of work-energy theorem and the conservation of energy.
1. First, let's calculate the initial kinetic energy (KE) of the block of ice:
KE_initial = (1/2) * mass * velocity^2
Given:
Mass of the block of ice (m) = 8.56 kg
Initial velocity (v_initial) = 15.6 m/s
KE_initial = (1/2) * 8.56 kg * (15.6 m/s)^2
2. Next, let's calculate the final kinetic energy (KE_final) of the block of ice after its speed has reduced to 10.2 m/s:
KE_final = (1/2) * mass * velocity^2
Given:
Final velocity (v_final) = 10.2 m/s
KE_final = (1/2) * 8.56 kg * (10.2 m/s)^2
3. The work done against friction can be calculated as the difference between the initial and final kinetic energies:
Work_done = KE_initial - KE_final
4. The heat generated during this process will be equal to the work done against friction, as stated in the problem:
Heat_generated = Work_done
5. To calculate the amount of ice that has melted, we need to determine the heat absorbed by the ice. Since half of the heat goes into the floor, the rest goes into melting the ice:
Heat_absorbed_by_ice = (1/2) * Heat_generated
6. Finally, we can calculate the amount of ice melted by using the specific latent heat of fusion of ice (L = 3.34e5 J/kg):
Mass_melted = Heat_absorbed_by_ice / L
Now let's plug in the given values and calculate the results:
KE_initial = (1/2) * 8.56 kg * (15.6 m/s)^2
KE_final = (1/2) * 8.56 kg * (10.2 m/s)^2
Work_done = KE_initial - KE_final
Heat_generated = Work_done
Heat_absorbed_by_ice = (1/2) * Heat_generated
Mass_melted = Heat_absorbed_by_ice / 3.34e5 J/kg
Calculating these values will give you the amount of ice melted after the speed of the ice has been reduced to 10.2 m/s. The maximum amount of ice that will melt can be determined by assuming that all the generated heat goes into melting the ice.