consider two solid cones made from a uniform material. The smaller cone is a ½ scale model of the bigger one. They are released from equal height, quickly reach their terminal velocities and fall in air until they hit the ground. Use scaling reasoning to predict which cone will reach the ground first
I know if air resistance is ignored it will be at the same time.
To predict which cone will reach the ground first, we can use scaling reasoning. Let's consider the important factors involved in determining the time of descent for each cone: mass and surface area.
Mass:
Since the smaller cone is a ½ scale model of the bigger one, its mass will be 1/8th of the bigger cone. This is because the volume of a cone is directly proportional to the cube of its dimensions. When we scale down the dimensions by a factor of 1/2, the volume (and consequently, the mass) scales down by a factor of (1/2)^3 = 1/8.
Surface Area:
The smaller cone will have a surface area that is 1/4th of the bigger cone's surface area. This is because surface area is directly proportional to the square of its dimensions. When we scale down the dimensions by a factor of 1/2, the surface area scales down by a factor of (1/2)^2 = 1/4.
Considering these scaling factors, we can conclude that the surface area of the smaller cone will be 1/4th, while its mass will be 1/8th compared to the bigger cone.
Now, let's consider the effect of air resistance. As you mentioned, if air resistance is ignored, both cones will reach the ground at the same time because their masses dictate the gravitational forces acting on them.
However, air resistance plays a crucial role in affecting the time of descent. When an object falls through the air, air resistance opposes its motion, slowing it down. The magnitude of air resistance depends on the surface area of the object. Thus, the cones' differing surface areas will cause a discrepancy in their times of descent.
Since the smaller cone has a smaller surface area, it will experience less air resistance compared to the bigger cone. As a result, the smaller cone will fall faster and reach the ground first.