There is an outcropping of ice with a large drop to a slope below.
A 100kg piece of ice breaks off with a 60kg person on top of it, and falls o the slope below. The piece of ice slides down the slope to the bottom and onto a horizontal portion.
It slides to a stop at a distance of 100m from the base of the slope. If the top of the slop is 650m vertically upward from the horizontal portion, and the block of ice is traveling 10 m/s when it contacts the slope, how much ice is melted during the trip? Make any reasonable assumptions to solve this problem.
I am not sure where to begin.. I understand we need to find the total distance traveled and to use the thermal conductivity of snow.. but i'm not sure where to start.. Thank you for your help in advance.
assumptions
... the person does not ride the ice down
... the height of the break is even with the top of the slope
the gravitational potential energy of the ice is converted to kinetic energy and then heat by friction
the total energy dissipated is
... m g h = 100 kg * g * 650 m
the heat of fusion of ice is
... 334 kJ/kg
divide the total energy by the heat of fusion to find the mass of ice melted
To solve this problem, we need to determine the distance traveled by the ice block, find the energy lost during the slide, and finally calculate the amount of melted ice.
1. Calculate the total distance traveled:
We are given that the top of the slope is 650m vertically above the horizontal portion, and the block slides to a stop at a distance of 100m from the base of the slope. This forms a right-angled triangle where the hypotenuse represents the total distance traveled by the ice block. We can use the Pythagorean theorem to find the hypotenuse distance.
Using the Pythagorean theorem:
Distance² = Vertical distance² + Horizontal distance²
Distance² = (650m)² + (100m)²
Distance² = 422,500m² + 10,000m²
Distance² = 432,500m²
Distance = √432,500m
Distance ≈ 657.7m
So, the total distance traveled by the ice block is approximately 657.7m.
2. Calculate the energy lost during the slide:
The energy lost during the slide can be calculated using the principle of conservation of mechanical energy. We need to find the difference between the initial mechanical energy (at the top of the slope) and the final mechanical energy (at the bottom of the slide). Since the ice block comes to a stop on the horizontal portion, its final kinetic energy is zero.
Initial mechanical energy = potential energy at the top
Final mechanical energy = potential energy at the bottom
Initial mechanical energy = mgh, where m is the mass (100kg), g is the acceleration due to gravity (9.8 m/s²), and h is the vertical height (650m).
Initial mechanical energy = 100kg x 9.8 m/s² x 650m
Final mechanical energy = mgh, where m is the mass (100kg), g is the acceleration due to gravity (9.8 m/s²), and h is the vertical height (0m).
Final mechanical energy = 100kg x 9.8 m/s² x 0m
Energy lost during the slide = Initial mechanical energy - Final mechanical energy
3. Calculate the amount of melted ice:
The amount of melted ice can be determined by converting the energy lost during the slide into heat energy and then dividing it by the thermal conductivity of ice.
Assuming the ice has a density of 917 kg/m³ and a specific heat capacity of 2,093 J/kg·K, we can use the equation:
Energy lost = mass of melted ice x specific heat capacity x temperature change
However, we need to find the mass of melted ice first.
The mass of melted ice can be found using the equation:
Energy lost = mass of melted ice x latent heat of fusion of ice
Where the latent heat of fusion of ice is approximately 333,500 J/kg.
Combining the two equations:
mass of melted ice = energy lost / latent heat of fusion of ice
Once we have the mass of melted ice, we can substitute it back into the first equation to calculate the amount of melted ice.
Please note that this explanation assumes no external forces, such as air resistance or friction, acting on the ice block during its slide. Additionally, these calculations make assumptions about the properties of ice.