Use dimensional analysis to convert 45 miles per hour to meters per second.

Step-by-step solution

1. 1

   Decide what has to cancel. The starting quantity has miles on top and hours on the bottom; the target has metres on top and seconds on the bottom. So you need one factor that puts miles in a denominator and one that puts hours in a numerator. Choosing the factors by which unit must cancel — not by which number looks right — is the whole idea of dimensional analysis.

2. 2

   Write the exact conversion factors. By definition

   $1\ \text{mi}=1609.344\ \text{m},\qquad 1\ \text{hr}=3600\ \text{s}$

   Both are exact, so no precision is lost. (The value $1609.344$ is exact because the international yard is defined as exactly $0.9144$ m; $1609.34$ is a rounded version.)

3. 3

   Chain the factors so the units cancel.

   $45\ \frac{\cancel{\text{mi}}}{\cancel{\text{hr}}}\times\frac{1609.344\ \text{m}}{1\ \cancel{\text{mi}}}\times\frac{1\ \cancel{\text{hr}}}{3600\ \text{s}}$

   Miles cancel against miles and hours against hours, leaving metres over seconds — the units of the answer confirm the setup before any arithmetic is done.

4. 4

   Do the arithmetic.

   $45\times\frac{1609.344}{3600}=45\times 0.44704=20.1168$

   so

   $45\ \text{mi/hr}=20.1168\ \text{m/s}\approx 20.12\ \text{m/s}$

   The grouped constant $0.44704$ is worth remembering: it converts any mi/hr figure to m/s in one multiplication.

5. 5

   Sanity-check the size and reuse the method. A mile is roughly $1.6$ km and an hour is $3600$ s, so mi/hr should shrink by a factor near $1609/3600\approx 0.45$ — and $45\times 0.45\approx 20$, matching. The same chaining works in reverse: $75\ \text{m/s}\times\frac{1\ \text{km}}{1000\ \text{m}}\times\frac{3600\ \text{s}}{1\ \text{hr}}=75\times 3.6=270\ \text{km/hr}$.