A highway curve forms a section of a circle. A car goes around the curve as shown in the helicopter view of Figure P1.48. Its dashboard compass shows that
the car is initially heading due east. After it travels 840 m, it is heading 35.0° south of east. Find the
radius of curvature of its path.
35° = 0.611 radians, so
.611r = 840
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A highway curve forms a section of a circle. A car goes around the curve. Its dashboard compass shows that the car is initially heading due east. After it travels 880. m, it is heading 10.0° south of east. Find the radius of curvature of its path.
To find the radius of curvature of the car's path, we can use trigonometry and the properties of circular motion.
Here's how you can approach it step by step:
1. Draw a diagram: Sketch a diagram of the situation. Label the initial position of the car as point A, the final position as point B, and the center of the curve as point O. Connect points A and B with a straight line segment to represent the 840 m distance traveled.
2. Define angles: Let angle θ be the angle formed between the initial direction (due east) and the final direction (south of east). In this case, we have θ = 35.0°.
3. Identify the triangle: In the diagram, you have a triangle formed by points A, B, and the center O. Since the car travels along a curved path, the straight line segment AB is tangent to the circle at point B.
4. Use trigonometry: In triangle AOB, use the properties of trigonometry to relate the angle θ to the radius of curvature. Specifically, you can use the tangent function:
tan(θ) = opposite/adjacent = AB/OB
Since AB = 840 m, we need to find OB.
5. Rearrange the equation: Rearrange the equation from step 4 to solve for OB:
OB = AB / tan(θ)
6. Calculate the radius: Calculate the value of tan(θ) using a scientific calculator or an online trigonometric function. Then, substitute this value into the equation from step 5 to find OB, which represents the radius of curvature of the car's path.
7. Evaluate the result: Once you have calculated the radius, make sure to check if it is a reasonable value given the context of the problem.
By following these steps, you should be able to find the radius of curvature of the car's path.