What happens if you separate the liquid from the melting ice (temperature wise)?
Yep, once the liquid water is not in contact with the solid ice, it will be able to more quickly equalize with the temperature of the air/container/etc that is around it. That means if you are in your kitchen and are melting a big ice cube, if you pour some of the melted water into a separate bowl, this water will reach room temperature quicker than the melted water that continues to accumulate under the ice cube.
