Problem 1-9: The distance moved in 1 sec and the speed in kilometers per million years of tectonic plates

Tectonic plates are large segments of the Earth’s crust that move slowly. Suppose that one such plate has an average speed of 4.0 cm/year.
(a) What distance does it move in 1.0 s at this speed?
(b) What is its speed in kilometers per million years?

We know that the formula for the distance traveled, d, if we are given rate, r, and time, t is given by $d=r\times t$ .

The rate in cm/s is

\begin{align*} r & =\left(4\:\frac{\text{cm}}{\bcancel{\text{year}}}\times \frac{1\:\bcancel{\text{year}}}{365\:\frac{1}{4}\:\bcancel{\text{days}}}\times \frac{1\:\bcancel{\text{day}}}{24\:\bcancel{\text{hours}}}\times \frac{1\:\bcancel{\text{hour}}}{60\:\bcancel{\text{mins}}}\times \frac{1\:\bcancel{\text{min}}}{60\:\text{s}}\right) \\ & = 1.2675 \times 10^{-7} \ \text{cm/s} \end{align*}

The distance traveled is

\begin{align*} d & = r \times t \\ d & = \left( 1.2675\times 10^{-7} \ \text{cm/s} \right)\left( 1.0 \ \text{s} \right) \\ d & = 1.3\times 10^{-7} \ \text{cm} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}

We need to convert 4.0 cm/year to km/My.

\begin{align*} 4.0 \ \frac{\text{cm}}{\text{year}} & = 4.0\:\frac{\bcancel{\text{cm}}}{\bcancel{\text{year}}}\times \frac{1\:\bcancel{\text{m}}}{100\:\bcancel{\text{cm}}}\times \frac{1\:\text{km}}{1000\:\bcancel{\text{m}}}\times \frac{1\:000\:000\:\bcancel{\text{years}}}{1\:\text{My}} \\ \\ & =40\:\text{km/My} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}

Therefore, the speed of the tectonic plates is 40 km/My.