## To calculate the height that a block of ice at 0°C must be dropped to completely melt upon impact, we can use the principle of conservation of energy.

The initial gravitational potential energy of the block at height h can be given by:

PE_initial = mgh

where m is the mass of the block, g is the acceleration due to gravity, and h is the height from which it is dropped.

When the block hits the ground, all of its initial gravitational potential energy is converted into heat energy to melt the ice.

The heat required to melt a certain mass of ice can be calculated using the specific heat capacity of ice (334 J/g°C) and the latent heat of fusion for ice (334,000 J/kg).

The heat required to completely melt the ice can be given by:

Q = m * c * ΔT + m * L_fusion

where Q is the heat energy, m is the mass of the ice, c is the specific heat capacity of ice, ΔT is the change in temperature, and L_fusion is the latent heat of fusion for ice.

For this problem, the change in temperature (ΔT) is 0°C since the ice is already at 0°C.

Let's assume the mass of the ice is 1 kg for simplicity.

Using the equation Q = m * c * ΔT + m * L_fusion and substituting the known values, we have:

Q = 1 kg * 334 J/g°C * 0°C + 1 kg * 334,000 J/kg

Simplifying this equation gives us:

Q = 0 J + 334,000 J

Now, equating the gravitational potential energy (PE_initial) to the heat energy (Q):

mgh = Q

Substituting the known values and solving for h:

1 kg * g * h = 334,000 J

Now, let's put in the value for the acceleration due to gravity (g), which is approximately 9.8 m/s²:

9.8 m/s² * h = 334,000 J

Solving for h:

h = 334,000 J / (9.8 m/s²)

h ≈ 34,082 m

Therefore, the block of ice must be dropped from a height of approximately 34,082 meters in order to completely melt upon impact, assuming all of its initial gravitational potential energy is used to melt the ice.