Consider two spherical stones with same density (mass per unit volume). The larger stone has radius R = 8.58 cm and the smaller stone has radius r = 3.37 cm. Find the ratio of the terminal speed in air of the larger stone to that of the smaller stone.
weight = k r^3
drag = c r^2 v^2
at terminal speed, drag/weight = 1
drag/weight = const v^2/r
so
V1^2/R1 = V2^2/R2
V2^2/V1^2 = R2/R1
V2/V1 = sqrt (R2/R1)
To find the ratio of the terminal speeds of the two stones, we need to consider the forces acting on them. At terminal speed, the force due to gravity is balanced by the force due to air resistance.
Let's start by calculating the weight of each stone. The weight formula is given by:
Weight = mass × gravitational acceleration
Since the stones have the same density, their mass will be proportional to their volume. The volume of a sphere is given by the formula:
Volume = (4/3) * π * radius^3
So, the mass of each stone will be:
mass = density × volume
Since both stones have the same density, we can ignore it for now. Let's calculate the weight of each stone.
For the larger stone with radius R = 8.58 cm, its weight will be:
Weight_R = mass_R × gravitational acceleration
For the smaller stone with radius r = 3.37 cm, its weight will be:
Weight_r = mass_r × gravitational acceleration
Now, let's calculate the terminal speed of each stone. At terminal speed, the force due to gravity is balanced by the force due to air resistance.
For the larger stone, the force due to gravity is given by its weight:
Force_gravity_R = Weight_R
For the smaller stone, the force due to gravity is given by its weight:
Force_gravity_r = Weight_r
The force due to air resistance is proportional to the velocity of the stone and acts in the opposite direction. At terminal speed, these forces are balanced:
Force_gravity_R = Force_air_resistance_R
Force_gravity_r = Force_air_resistance_r
Now, let's calculate the ratio of the terminal speeds:
Ratio = velocity_R / velocity_r
To find the velocity of each stone, we need to solve the equations:
Weight_R = Force_air_resistance_R
Weight_r = Force_air_resistance_r
Once we find the forces due to air resistance, we can use Newton's Second Law (Force = mass × acceleration) to find the acceleration of each stone. Finally, we can integrate the acceleration over time to find the terminal velocities.
The calculation involves several steps and may require numerical methods, so it would be best to use a physics simulation software or consult a physics expert for an accurate result.