Answer :
To solve this problem, we need to find the coefficient of friction between the road and the tires of a car that skids to a stop. Here's how we can do it step-by-step:
1. Understand the Problem:
- A car weighing 8500 N (which is the force due to gravity on the car) is initially traveling at 20 meters per second (m/s).
- The car comes to a stop after skidding 200 meters.
- We need to find the coefficient of friction (µ) between the car's tires and the road.
2. Apply the Work-Energy Principle:
- The work done by friction is equal to the change in kinetic energy of the car.
- The initial kinetic energy ([tex]\(KE_{\text{initial}}\)[/tex]) of the car is given by the formula:
[tex]\[
KE_{\text{initial}} = \frac{1}{2} \times m \times v^2
\][/tex]
- Here, [tex]\(m\)[/tex] is the mass of the car, which we find from the weight ([tex]\(W = mg\)[/tex]) by rearranging:
[tex]\[
m = \frac{W}{g} = \frac{8500 \, \text{N}}{9.81 \, \text{m/s}^2}
\][/tex]
3. Calculate Work Done by Friction:
- The work done by friction, [tex]\(W_{\text{friction}}\)[/tex], is equal to the frictional force ([tex]\(f_{\text{friction}}\)[/tex]) times the distance ([tex]\(d\)[/tex]) the car skidded:
[tex]\[
W_{\text{friction}} = f_{\text{friction}} \times d
\][/tex]
- Since the entire initial kinetic energy is dissipated through friction:
[tex]\[
W_{\text{friction}} = KE_{\text{initial}}
\][/tex]
4. Solve for Frictional Force:
- The frictional force is related to the normal force ([tex]\(N\)[/tex]) and the coefficient of friction ([tex]\(µ\)[/tex]):
[tex]\[
f_{\text{friction}} = µ \times N
\][/tex]
- Here, the normal force ([tex]\(N\)[/tex]) equals the weight of the car (since there are no vertical accelerations), [tex]\(N = W = 8500 \, \text{N}\)[/tex].
5. Find the Coefficient of Friction:
- Rearrange the formula to solve for the coefficient of friction ([tex]\(µ\)[/tex]):
[tex]\[
µ = \frac{W_{\text{friction}}}{N \times d}
\][/tex]
[tex]\[
µ = \frac{KE_{\text{initial}}}{W \times d}
\][/tex]
Given the calculations, the coefficient of friction is approximately 0.1. Based on the options provided:
- B) 0.2
- C) 0.5
- D) 0.8
None of these options are a perfect match. However, given the closest option, you might consider checking approximate or rounded values depending on the method of calculation or the assumptions made. But strictly interpreting the calculations, none of the choices match exactly, pointing to a potential additional consideration in interpreting the problem's choices or a typographical error in the options.
1. Understand the Problem:
- A car weighing 8500 N (which is the force due to gravity on the car) is initially traveling at 20 meters per second (m/s).
- The car comes to a stop after skidding 200 meters.
- We need to find the coefficient of friction (µ) between the car's tires and the road.
2. Apply the Work-Energy Principle:
- The work done by friction is equal to the change in kinetic energy of the car.
- The initial kinetic energy ([tex]\(KE_{\text{initial}}\)[/tex]) of the car is given by the formula:
[tex]\[
KE_{\text{initial}} = \frac{1}{2} \times m \times v^2
\][/tex]
- Here, [tex]\(m\)[/tex] is the mass of the car, which we find from the weight ([tex]\(W = mg\)[/tex]) by rearranging:
[tex]\[
m = \frac{W}{g} = \frac{8500 \, \text{N}}{9.81 \, \text{m/s}^2}
\][/tex]
3. Calculate Work Done by Friction:
- The work done by friction, [tex]\(W_{\text{friction}}\)[/tex], is equal to the frictional force ([tex]\(f_{\text{friction}}\)[/tex]) times the distance ([tex]\(d\)[/tex]) the car skidded:
[tex]\[
W_{\text{friction}} = f_{\text{friction}} \times d
\][/tex]
- Since the entire initial kinetic energy is dissipated through friction:
[tex]\[
W_{\text{friction}} = KE_{\text{initial}}
\][/tex]
4. Solve for Frictional Force:
- The frictional force is related to the normal force ([tex]\(N\)[/tex]) and the coefficient of friction ([tex]\(µ\)[/tex]):
[tex]\[
f_{\text{friction}} = µ \times N
\][/tex]
- Here, the normal force ([tex]\(N\)[/tex]) equals the weight of the car (since there are no vertical accelerations), [tex]\(N = W = 8500 \, \text{N}\)[/tex].
5. Find the Coefficient of Friction:
- Rearrange the formula to solve for the coefficient of friction ([tex]\(µ\)[/tex]):
[tex]\[
µ = \frac{W_{\text{friction}}}{N \times d}
\][/tex]
[tex]\[
µ = \frac{KE_{\text{initial}}}{W \times d}
\][/tex]
Given the calculations, the coefficient of friction is approximately 0.1. Based on the options provided:
- B) 0.2
- C) 0.5
- D) 0.8
None of these options are a perfect match. However, given the closest option, you might consider checking approximate or rounded values depending on the method of calculation or the assumptions made. But strictly interpreting the calculations, none of the choices match exactly, pointing to a potential additional consideration in interpreting the problem's choices or a typographical error in the options.
