Discover the answers you need at Westonci.ca, where experts provide clear and concise information on various topics. Explore thousands of questions and answers from a knowledgeable community of experts on our user-friendly platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
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.
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.
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
