The problem asks us to convert $32$ m/s to mph using the given conversion factor $8$ km/h $= 5$ mph.

1. Problem Description

The problem asks us to convert $32$ m/s to mph using the given conversion factor $8$ km/h $= 5$ mph.

2. Solution Steps

First, convert $32$ m/s to km/h.

We know that $1$ km $= 1000$ m, and $1$ hour $= 3600$ seconds.

So, $1$ m/s $= (1/1000)$ km / $(1/3600)$ h $= (3600/1000)$ km/h $= 3.6$ km/h.

Therefore, $32$ m/s $= 32 \times 3.6$ km/h $= 115.2$ km/h.

Now, use the given conversion $8$ km/h $= 5$ mph to convert $115.2$ km/h to mph.

We can write the conversion factor as $\frac{5 \text{ mph}}{8 \text{ km/h}}$.

Multiply $115.2$ km/h by this conversion factor:

$115.2 \text{ km/h} \times \frac{5 \text{ mph}}{8 \text{ km/h}} = \frac{115.2 \times 5}{8} \text{ mph} = \frac{576}{8} \text{ mph} = 72 \text{ mph}$.

3. Final Answer

The final answer is 72 mph.