To determine the stopping distance of a car that weighs 4500 pounds and is traveling at 65 miles per hour when it is braked suddenly with a constant braking force of 1100 pounds, let's follow these steps:
1. Convert the initial velocity from miles per hour to feet per second:
[tex]\[
1 \text{ mile per hour (mi/h)} = 1.467 \text{ feet per second (ft/s)}
\][/tex]
Thus,
[tex]\[
65 \text{ mi/h} \times 1.467 = 95.355 \text{ ft/s}
\][/tex]
2. Convert the weight of the car to mass in slugs:
The weight (force) of the car is given as 4500 pounds. The standard acceleration due to gravity is approximately 32.174 feet per second squared (ft/s²). The relationship between weight, mass, and gravity is given by:
[tex]\[
\text{mass} = \frac{\text{weight}}{\text{gravity}}
\][/tex]
Therefore,
[tex]\[
\text{mass} = \frac{4500 \text{ lb}}{32.174 \text{ ft/s}^2} \approx 139.874 \text{ slugs}
\][/tex]
3. Calculate the initial kinetic energy of the car:
The kinetic energy (KE) of an object is given by:
[tex]\[
\text{KE} = \frac{1}{2} \times \text{mass} \times \text{velocity}^2
\][/tex]
So for the car,
[tex]\[
\text{KE} = \frac{1}{2} \times 139.874 \text{ slugs} \times (95.355 \text{ ft/s})^2 \approx 634614.333 \text{ foot-pounds}
\][/tex]
4. Compute the stopping distance using the work-energy principle:
The work done by the braking force in bringing the car to a stop must equal the initial kinetic energy of the car. Mathematically,
[tex]\[
\text{Work} = \text{Braking force} \times \text{distance}
\][/tex]
This means,
[tex]\[
\text{distance} = \frac{\text{KE}}{\text{Braking force}} = \frac{634614.333}{1100} \approx 576.013 \text{ feet}
\][/tex]
5. Convert the stopping distance from feet to yards:
There are 3 feet in a yard. Thus,
[tex]\[
\text{distance in yards} = \frac{576.013 \text{ feet}}{3} \approx 192.671 \text{ yards}
\][/tex]
6. Select the closest option from the given choices:
The options provided are 110 yards, 130 yards, 170 yards, 190 yards, and 240 yards. The closest match to our calculated distance of approximately 192.671 yards is:
[tex]\[
190 \text{ yards}
\][/tex]
Therefore, the answer is [tex]\( \boxed{190} \)[/tex] yards.
