So-called volcanic 'ash' is actually finely pulverized rock blown high into the atmosphere. A typical ash particle is a 50-μm-diameter piece of silica with a density of 2400 kg/m³. How long would it take this ash particle to fall from a height of 5.0 km in vacuum?

Verified step by step guidance

1. Determine the mass of the ash particle using its volume and density. The volume of a sphere is given by \( V = \frac{4}{3} \pi r^3 \), where \( r \) is the radius of the particle. The radius is half the diameter, so \( r = \frac{50 \times 10^{-6}}{2} \) meters. Then, calculate the mass using \( m = \rho V \), where \( \rho \) is the density (2400 kg/m^3).
2. In a vacuum, the only force acting on the particle is gravity. The acceleration due to gravity is \( g = 9.8 \ \text{m/s}^2 \). The motion of the particle can be treated as free fall under constant acceleration.
3. Use the kinematic equation \( h = \frac{1}{2} g t^2 \) to solve for the time \( t \). Here, \( h \) is the height (5.0 km or 5000 m), \( g \) is the acceleration due to gravity, and \( t \) is the time to fall.
4. Rearrange the equation to solve for \( t \): \( t = \sqrt{\frac{2h}{g}} \). Substitute \( h = 5000 \ \text{m} \) and \( g = 9.8 \ \text{m/s}^2 \) into the equation.
5. Perform the calculation to find the time \( t \). This will give the time it takes for the ash particle to fall from a height of 5.0 km in a vacuum.

Key Concepts

Free Fall

Free fall refers to the motion of an object under the influence of gravity alone, without any air resistance. In a vacuum, all objects fall at the same rate regardless of their mass, which is approximately 9.81 m/s² on Earth. This principle allows us to calculate the time it takes for an object to fall from a certain height using the equations of motion.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. For free fall, the relevant equation is h = 0.5 * g * t², where h is the height, g is the acceleration due to gravity, and t is the time of fall. This equation can be rearranged to solve for time when the height and acceleration are known.

Density and Volume

Density is defined as mass per unit volume and is crucial for understanding the properties of materials. In this context, the density of the ash particle (2400 kg/m³) helps to identify its mass if the volume is known. However, in a vacuum scenario, the density does not affect the fall time, as all objects fall at the same rate regardless of their mass or density.