So-called volcanic 'ash' is actually finely pulverized rock blown high into the atmosphere. A typical ash particle is a 50-micrometer-diameter piece of silica with a density of 2400 kg/m^3. (b) 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. Calculate the terminal velocity of the ash particle. Terminal velocity occurs when the force of gravity on the particle is balanced by the drag force of the air. Use the formula for terminal velocity, $v_t = \sqrt{\frac{2mg}{\rho C_d A}}$, where $m$ is the mass of the particle, $g$ is the acceleration due to gravity (approximately 9.81 m/s^2), $\rho$ is the density of air (approximately 1.2 kg/m^3 at 20°C and sea level), $C_d$ is the drag coefficient (typically around 0.5 for spherical particles), and $A$ is the cross-sectional area of the particle ($A = \pi r^2$, where $r$ is the radius of the particle).
2. Calculate the mass of the ash particle using its density and volume. The volume of a sphere is given by $V = \frac{4}{3}\pi r^3$, and the mass is then $m = \rho_{\text{particle}} V$, where $\rho_{\text{particle}}$ is the density of the particle (2400 kg/m^3).
3. Calculate the cross-sectional area of the particle. Since the particle is spherical, its cross-sectional area $A$ can be calculated using the formula $A = \pi r^2$, where $r$ is the radius of the particle. Convert the diameter to radius by dividing by 2.
4. Substitute the values of $m$, $g$, $\rho$, $C_d$, and $A$ into the terminal velocity formula to find $v_t$.
5. Calculate the time it takes for the ash particle to fall from a height of 5.0 km using the formula $t = \frac{h}{v_t}$, where $h$ is the height (5.0 km) and $v_t$ is the terminal velocity. Convert this time into hours.

Key Concepts

Terminal Velocity

Terminal velocity is the constant speed an object reaches when the force of gravity pulling it down is balanced by the drag force acting against it. For small particles like volcanic ash, this speed is influenced by their size, shape, and the density of the fluid (air) they are falling through. Understanding terminal velocity is crucial for calculating how long it takes for the ash to fall from a specific height.

Drag Force

The drag force is the resistance experienced by an object moving through a fluid, such as air. It depends on factors like the object's velocity, cross-sectional area, and the fluid's density and viscosity. In the case of volcanic ash, the drag force plays a significant role in determining how quickly the ash particle descends through the atmosphere, especially at lower velocities.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. They relate displacement, initial velocity, final velocity, acceleration, and time. In this scenario, these equations can be used to calculate the time it takes for the ash particle to fall from 5.0 km, considering the effects of gravity and drag force until it reaches terminal velocity.