So-called volcanic 'ash' is actually finely pulverized rock blown high into the atmosphere. A typical ash particle is a 50-$\mu m$-diameter piece of silica with a density of 2400 kg/m³. How long in hours does it take this ash particle to fall from a height of 5.0 km in still air? Use the properties of 20°C air at sea level.

Verified step by step guidance

1. Step 1: Identify the forces acting on the ash particle. The primary forces are gravity pulling the particle downward and air resistance opposing the motion. The particle will reach terminal velocity when these forces balance.
2. Step 2: Calculate the terminal velocity of the ash particle. Use the formula for terminal velocity: $v$_t=2mgρAC_d, where $m$ is the mass of the particle, $g$ is the acceleration due to gravity, $\rho$ is the density of air, $A$ is the cross-sectional area of the particle, and $C$_d is the drag coefficient.
3. Step 3: Determine the mass of the ash particle. Use the formula for the volume of a sphere: $V=\frac{4}{3}\pi r^3$, where $r$ is the radius of the particle. Multiply the volume by the density of silica to find the mass.
4. Step 4: Calculate the cross-sectional area of the ash particle. Use the formula for the area of a circle: $A=\pi r^2$, where $r$ is the radius of the particle.
5. Step 5: Once the terminal velocity is calculated, determine the time it takes for the particle to fall from a height of 5.0 km. Use the formula $t=\frac{h}{v}$_t, where $h$ is the height and $v$_t is the terminal velocity. Convert the time from seconds to hours.

Key Concepts

Terminal Velocity

Terminal velocity is the constant speed that a freely falling object eventually reaches when the resistance of the medium through which it is falling prevents further acceleration. For small particles like volcanic ash, this occurs when the force of gravity is balanced by the drag force due to air resistance. Understanding terminal velocity is crucial for calculating how long it takes for the ash particle to fall from a specific height.

Drag Force

The drag force is the resistance force experienced by an object moving through a fluid, such as air. It depends on the object's velocity, cross-sectional area, and the fluid's density. For small particles like volcanic ash, the drag force can be modeled using Stokes' law, which is particularly relevant at low Reynolds numbers, where the flow is laminar and the particle size is small compared to the fluid's viscosity.

Free Fall and Gravity

Free fall refers to the motion of an object under the influence of gravitational force alone, without any other forces acting on it. On Earth, the acceleration due to gravity is approximately 9.81 m/s². In the context of the ash particle, understanding free fall is essential to determine how gravity affects its descent and how long it will take to reach the ground from a height of 5.0 km.