To determine the speed of the vehicle at the moment the driver hit the brakes, we use the given formula:
[tex]\[ S=\sqrt{30 D \cdot f \cdot n} \][/tex]
Here’s a step-by-step explanation of how to use this formula:
1. Identify and Understand Variables:
- [tex]\( S \)[/tex] represents the speed of the vehicle in miles per hour (mph).
- [tex]\( 30 \)[/tex] is a constant that helps in converting the units properly (from feet and hours to miles per hour).
- [tex]\( D \)[/tex] is the skid distance measured in feet.
- [tex]\( f \)[/tex] is the drag factor, which varies depending on the road surface (e.g., asphalt, gravel, etc.).
- [tex]\( n \)[/tex] is an adjustment factor representing the number of skid marks or tires involved.
2. Plug in the Values:
For this problem, the example values are:
- [tex]\( D = 100 \)[/tex] feet (skid distance)
- [tex]\( f = 0.75 \)[/tex] (drag factor for the given road surface)
- [tex]\( n = 1 \)[/tex] (adjustment factor for a single skid mark)
3. Substitute the Values into the Formula:
[tex]\[
S = \sqrt{30 \cdot 100 \cdot 0.75 \cdot 1}
\][/tex]
4. Simplify the Expression Under the Square Root:
- First, calculate [tex]\( 30 \cdot 100 \)[/tex]:
[tex]\[
30 \cdot 100 = 3000
\][/tex]
- Then, multiply this result by 0.75:
[tex]\[
3000 \cdot 0.75 = 2250
\][/tex]
5. Take the Square Root:
[tex]\[
S = \sqrt{2250}
\][/tex]
6. Calculate the Square Root:
By finding the square root of 2250, we get:
[tex]\[
S \approx 47.43416490252569
\][/tex]
Therefore, the estimated speed of the vehicle is approximately [tex]\( 47.434 \)[/tex] miles per hour at the moment the driver applied the brakes.
