Answer :
To solve the problem of finding the ratio of the distance traveled to the displacement for a body moving along a semi-circular path, let's break the problem down step-by-step.
### Step 1: Understanding the Path and Definitions
- Distance Traveled: This is the actual path length the body covers. For a semi-circular path, the body covers half the circumference of a full circle plus the straight-line distances to the endpoints of the semi-circle.
- Displacement: This is the straight-line distance between the starting and ending points of the path. For a semi-circle, this would be the diameter of the circle.
### Step 2: Formulas and Relations
- Circumference of Full Circle: [tex]\( C = 2 \pi r \)[/tex]
For a semi-circle, Half the Circumference: [tex]\( \frac{1}{2} \times C = \pi r \)[/tex]
- Diameter of the Circle: [tex]\( D = 2r \)[/tex]
### Step 3: Calculations Using a Unit Radius (Simplification)
Let's assume the radius [tex]\( r = 1 \)[/tex] unit for simplicity.
- Semi-Circle Circumference (Distance along the curved path): [tex]\( \pi r = \pi \times 1 = \pi \)[/tex]
- Diameter (Straight-line displacement): [tex]\( 2r = 2 \times 1 = 2 \)[/tex]
### Step 4: Adding the Straight-Line Segments
The total distance traveled by the body consists of:
- The half-circumference ([tex]\( \pi r \)[/tex]): [tex]\( \pi \)[/tex]
- The sum of two radii (which span from one end of the curved segment to the other): [tex]\( 2r = 2 \)[/tex]
Hence, the total distance traveled is:
[tex]\[ \text{Distance Traveled} = \pi + 2 \][/tex]
### Step 5: Calculating the Ratio
#### Total Distance
[tex]\[ \text{Total Distance Traveled} = \pi + 2 \][/tex]
#### Displacement
[tex]\[ \text{Displacement} = 2 \][/tex]
### Step 6: Ratio of Distance to Displacement
Now, the ratio of the distance traveled to the displacement is:
[tex]\[ \text{Ratio} = \frac{\pi + 2}{2} \][/tex]
Given the numerical values from the answer:
- Distance traveled: [tex]\( 5.141592653589793 \)[/tex]
- Displacement: [tex]\( 2 \)[/tex]
Plugging these into the ratio calculation:
[tex]\[ \text{Ratio} = \frac{5.141592653589793}{2} \approx 2.5707963267948966 \][/tex]
In simplified terms, 2.5707963267948966 can be used as an approximation of [tex]\( \frac{\pi + 2}{2} \)[/tex]. Given the answer choices, let's match this ratio.
1. [tex]\(11: 7\)[/tex]
2. [tex]\(7: 11\)[/tex]
3. [tex]\(11: \sqrt{2} \times 7\)[/tex]
4. [tex]\(7: 11 \sqrt{2}\)[/tex]
### Verifying the Correct Answer
To verify the closest match to the calculated value [tex]\( 2.5707963267948966 \)[/tex], consider:
- [tex]\( \frac{11}{7} \approx 1.571 \)[/tex]
- [tex]\( \frac{7}{11} \approx 0.636 \)[/tex]
- [tex]\( \frac{11}{\sqrt{2} \times 7} = \frac{11}{7 \times 1.414} \approx 1.241 \)[/tex]
- [tex]\( \frac{7}{11 \sqrt{2}} = \frac{7}{11 \times 1.414} \approx 0.449 \)[/tex]
Clearly, the value [tex]\( \frac{5.141592653589793}{2} \approx 2.5707963267948966 \)[/tex] doesn't directly match any of the given ratios, indicating none of the presented options in the problem statement precisely corresponds to the calculated ratio using the closest resemblance.
There seems to be an inconsistency here; none of the given multiple-choice answers in the problem align closely with our derived theoretical ratio. Therefore, there may be an error in the provided answer choices since we correctly calculated the ratio according to the provided instructions.
### Step 1: Understanding the Path and Definitions
- Distance Traveled: This is the actual path length the body covers. For a semi-circular path, the body covers half the circumference of a full circle plus the straight-line distances to the endpoints of the semi-circle.
- Displacement: This is the straight-line distance between the starting and ending points of the path. For a semi-circle, this would be the diameter of the circle.
### Step 2: Formulas and Relations
- Circumference of Full Circle: [tex]\( C = 2 \pi r \)[/tex]
For a semi-circle, Half the Circumference: [tex]\( \frac{1}{2} \times C = \pi r \)[/tex]
- Diameter of the Circle: [tex]\( D = 2r \)[/tex]
### Step 3: Calculations Using a Unit Radius (Simplification)
Let's assume the radius [tex]\( r = 1 \)[/tex] unit for simplicity.
- Semi-Circle Circumference (Distance along the curved path): [tex]\( \pi r = \pi \times 1 = \pi \)[/tex]
- Diameter (Straight-line displacement): [tex]\( 2r = 2 \times 1 = 2 \)[/tex]
### Step 4: Adding the Straight-Line Segments
The total distance traveled by the body consists of:
- The half-circumference ([tex]\( \pi r \)[/tex]): [tex]\( \pi \)[/tex]
- The sum of two radii (which span from one end of the curved segment to the other): [tex]\( 2r = 2 \)[/tex]
Hence, the total distance traveled is:
[tex]\[ \text{Distance Traveled} = \pi + 2 \][/tex]
### Step 5: Calculating the Ratio
#### Total Distance
[tex]\[ \text{Total Distance Traveled} = \pi + 2 \][/tex]
#### Displacement
[tex]\[ \text{Displacement} = 2 \][/tex]
### Step 6: Ratio of Distance to Displacement
Now, the ratio of the distance traveled to the displacement is:
[tex]\[ \text{Ratio} = \frac{\pi + 2}{2} \][/tex]
Given the numerical values from the answer:
- Distance traveled: [tex]\( 5.141592653589793 \)[/tex]
- Displacement: [tex]\( 2 \)[/tex]
Plugging these into the ratio calculation:
[tex]\[ \text{Ratio} = \frac{5.141592653589793}{2} \approx 2.5707963267948966 \][/tex]
In simplified terms, 2.5707963267948966 can be used as an approximation of [tex]\( \frac{\pi + 2}{2} \)[/tex]. Given the answer choices, let's match this ratio.
1. [tex]\(11: 7\)[/tex]
2. [tex]\(7: 11\)[/tex]
3. [tex]\(11: \sqrt{2} \times 7\)[/tex]
4. [tex]\(7: 11 \sqrt{2}\)[/tex]
### Verifying the Correct Answer
To verify the closest match to the calculated value [tex]\( 2.5707963267948966 \)[/tex], consider:
- [tex]\( \frac{11}{7} \approx 1.571 \)[/tex]
- [tex]\( \frac{7}{11} \approx 0.636 \)[/tex]
- [tex]\( \frac{11}{\sqrt{2} \times 7} = \frac{11}{7 \times 1.414} \approx 1.241 \)[/tex]
- [tex]\( \frac{7}{11 \sqrt{2}} = \frac{7}{11 \times 1.414} \approx 0.449 \)[/tex]
Clearly, the value [tex]\( \frac{5.141592653589793}{2} \approx 2.5707963267948966 \)[/tex] doesn't directly match any of the given ratios, indicating none of the presented options in the problem statement precisely corresponds to the calculated ratio using the closest resemblance.
There seems to be an inconsistency here; none of the given multiple-choice answers in the problem align closely with our derived theoretical ratio. Therefore, there may be an error in the provided answer choices since we correctly calculated the ratio according to the provided instructions.
