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Sagot :
To solve the problem of finding the ratio of the distance traveled to the displacement of a body moving along the curved path of a semi-circle, let's go through the detailed steps:
1. Understanding the Problem:
- The body moves along a semi-circular path.
- We need to calculate the distance the body travels along the semi-circle and its displacement.
- Then, we find the ratio of the distance to the displacement.
2. Given Information:
- The radius of the semi-circle is 7 units (this is implicitly known).
3. Calculations:
- Distance traveled (Curved Path):
- The distance along the semi-circle is half the circumference of the full circle.
- The circumference of the full circle = [tex]\( 2 \pi \times \text{radius} \)[/tex]
- Therefore, the circumference of the semi-circle (distance) = [tex]\( \pi \times \text{radius} \)[/tex]
- With radius = 7 units, the distance = [tex]\( \pi \times 7 \)[/tex]
- Numerical value: ~21.991
- Displacement (Straight-Line Distance):
- The displacement is the straight-line distance between the starting and ending points of the semi-circle, which is the diameter.
- The diameter of the semi-circle = 2 \times \text{radius}
- With radius = 7 units, displacement = 2 \times 7 = 14 units
4. Ratio of Distance to Displacement:
- Ratio = [tex]\(\frac{\text{Distance}}{\text{Displacement}}\)[/tex]
- Numerical value: ~1.5708
5. Comparing with Given Options:
- Let's simplify the options in numerical form:
- Option (1) [tex]\( \frac{11}{7} \approx 1.5714 \)[/tex] (close to our ratio)
- Option (2) [tex]\( \frac{7}{11} \approx 0.636 \)[/tex] (not close)
- Option (3) [tex]\( \frac{11}{7\sqrt{2}} \)[/tex]
- Simplifying, [tex]\( \sqrt{2} \approx 1.414 \)[/tex]
- [tex]\( 7\sqrt{2} \approx 9.899 \)[/tex]
- [tex]\( \frac{11}{9.899} \approx 1.112 \)[/tex] (not close)
- Option (4) [tex]\( \frac{7}{11\sqrt{2}} \)[/tex]
- Simplify [tex]\( 11\sqrt{2} \approx 15.556 \)[/tex]
- [tex]\( \frac{7}{15.556} \approx 0.45 \)[/tex] (not close)
The closest match to our found value of ~1.5708 is given by option (1):
[tex]\[ \boxed{11:7} \][/tex]
1. Understanding the Problem:
- The body moves along a semi-circular path.
- We need to calculate the distance the body travels along the semi-circle and its displacement.
- Then, we find the ratio of the distance to the displacement.
2. Given Information:
- The radius of the semi-circle is 7 units (this is implicitly known).
3. Calculations:
- Distance traveled (Curved Path):
- The distance along the semi-circle is half the circumference of the full circle.
- The circumference of the full circle = [tex]\( 2 \pi \times \text{radius} \)[/tex]
- Therefore, the circumference of the semi-circle (distance) = [tex]\( \pi \times \text{radius} \)[/tex]
- With radius = 7 units, the distance = [tex]\( \pi \times 7 \)[/tex]
- Numerical value: ~21.991
- Displacement (Straight-Line Distance):
- The displacement is the straight-line distance between the starting and ending points of the semi-circle, which is the diameter.
- The diameter of the semi-circle = 2 \times \text{radius}
- With radius = 7 units, displacement = 2 \times 7 = 14 units
4. Ratio of Distance to Displacement:
- Ratio = [tex]\(\frac{\text{Distance}}{\text{Displacement}}\)[/tex]
- Numerical value: ~1.5708
5. Comparing with Given Options:
- Let's simplify the options in numerical form:
- Option (1) [tex]\( \frac{11}{7} \approx 1.5714 \)[/tex] (close to our ratio)
- Option (2) [tex]\( \frac{7}{11} \approx 0.636 \)[/tex] (not close)
- Option (3) [tex]\( \frac{11}{7\sqrt{2}} \)[/tex]
- Simplifying, [tex]\( \sqrt{2} \approx 1.414 \)[/tex]
- [tex]\( 7\sqrt{2} \approx 9.899 \)[/tex]
- [tex]\( \frac{11}{9.899} \approx 1.112 \)[/tex] (not close)
- Option (4) [tex]\( \frac{7}{11\sqrt{2}} \)[/tex]
- Simplify [tex]\( 11\sqrt{2} \approx 15.556 \)[/tex]
- [tex]\( \frac{7}{15.556} \approx 0.45 \)[/tex] (not close)
The closest match to our found value of ~1.5708 is given by option (1):
[tex]\[ \boxed{11:7} \][/tex]
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