Question

If one micrometeorite (a sphere with a diameter of 1.4 10-6 m) struck each square meter of the Moon each second, estimate the number of years it would take to cover the Moon with micrometeorites to a depth of one meter? Hint: Consider the volume of a micrometeorite and a cubic box, 1 m on a side, on the Moon. Find how long it would take to fill the box. (Type your answer using one of the following formats, 1.2e-3 for 0.0012 and 1.20e+2 for 120)

Answer 1

Volume of box is equal to net volume of micrometeorites.\[V_b = \sum_i V_{mi}\]Since all micrometeorites are assumed to be of the same volume, for some constant C, we can say\[V_b = CV_m\]\[C = \dfrac{V_b}{V_m}\]Plug in known values:\[C = \dfrac{1 \text{ meter}^3}{1.4 \times 10^{-6}\dfrac{\text{ meters}^3}{\text{micrometeorite}}}\]\[C=\dfrac{5}{7}\times 10^6\text{ meteors}\]Use the given and otherwise known conversion factors to convert micrometeorites to years.\[C=\left(\dfrac{5}{7}\times 10^6\text{ micrometeorites}\right)\left(\dfrac{1\text{ second}}{1\text{ micrometeorite}}\right)\left(\dfrac{1\text{ minute}}{60 \text{ seconds}}\right)\left(\dfrac{1 \text{ hour}}{60 \text{ minutes}}\right)\left(\dfrac{1\text{ day}}{24\text{ hours}}\right)\left(1\dfrac{\text{ year}}{365\text{ days}}\right)\]I'll leave it to you to evaluate that. :P