Problem 71 - College Physics

Question

In the Tour de France, a bicyclist races up two successive (straight) hills of different slope and length. The first is \(2.00 \mathrm{~km}\) long at an angle of \(5^{\circ}\) above the horizontal. This is immediately followed by one \(3.00 \mathrm{~km}\) long at \(7^{\circ}\). (a) What will be the overall (net) angle from start to finish: (1) smaller than \(5^{\circ},\) (2) between \(5^{\circ}\) and \(7^{\circ}\), or (3) greater than \(7^{\circ} ?\) (b) Calculate the actual overall (net) angle of rise experienced by this racer from start to finish, to corroborate your reasoning in part (a).

Step-by-Step Solution

Answer

(b) The net angle is calculated by finding the arctangent of the total vertical rise over the total horizontal distance.

Step 1: Calculate Vertical Rise of Each Hill

Start by calculating the vertical rise for each hill. The vertical rise for the first hill can be found using the sine function: \[ \text{Vertical rise for first hill} = 2.00 \text{ km} \times \sin(5^{\circ}) \]Similarly, the vertical rise for the second hill is:\[ \text{Vertical rise for second hill} = 3.00 \text{ km} \times \sin(7^{\circ}) \]

Step 2: Find Total Vertical Rise

Add the vertical rises of both hills to get the total vertical rise:\[ \text{Total vertical rise} = \left(2.00 \times \sin(5^{\circ})\right) + \left(3.00 \times \sin(7^{\circ})\right) \]

Step 3: Calculate Total Horizontal Distance

Compute the total horizontal component of both hills using the cosine function:\[ \text{Horizontal distance} = 2.00 \times \cos(5^{\circ}) + 3.00 \times \cos(7^{\circ}) \]

Step 4: Determine Net Angle of Elevation

Use the tangent function to find the net angle of elevation. \[ \text{Net angle} = \arctan\left(\frac{\text{Total vertical rise}}{\text{Total horizontal distance}}\right) \]Calculate the arctangent to find the net angle.

Key Concepts

Trigonometry in Physics

When tackling physics problems, trigonometry is often an invaluable tool, especially when dealing with slopes and angles. The key functions in trigonometry—sine, cosine, and tangent—help us relate angles to the sides of a right triangle. This makes them extremely useful in calculating distances and angles in physics.
These functions can help break down vector quantities like slopes or forces into components that are easier to work with. Here, physics problems that involve calculating angles of elevation or descent on a slope, often depend on these trigonometric functions.

• The **sine** function (\(\sin \theta\)) relates the angle to the opposite side over the hypotenuse.
• The **cosine** function (\(\cos \theta\)) relates the angle to the adjacent side over the hypotenuse.
• The **tangent** function (\(\tan \theta\)) is used to find the ratio of the opposite side to the adjacent side.

In our cycling example, trigonometry helps us determine how much the elevation changes and how far the cyclist travels horizontally. By understanding these functions, you can better solve complex problems that involve different slopes.

Net Angle of Elevation

The net angle of elevation is an important concept in problems involving slopes or inclines because it represents the single overall angle that best describes the entire journey.
This angle provides a simplified perspective on a complex series of movements. To calculate the net angle of elevation, we make use of the vertical and horizontal components of each segment of a path.
In our exercise, each hill has a vertical rise and a horizontal distance. By combining these two sets of data, we can find a single angle that captures the average inclination of the entire route.

• First, calculate each individual vertical rise using their respective angles (as seen with the sine function).
• Then, find each horizontal component using the cosine function.
• Add up the total vertical and horizontal measures.
• Finally, apply the tangent function to find the angle using \(\arctan\left(\frac{\text{Total vertical rise}}{\text{Total horizontal distance}}\right)\).

This net angle not only completes the exercise but gives insights into the efficiency of the route.

Vertical and Horizontal Components

Breaking down a motion into vertical and horizontal components is a fundamental method in physics problem-solving. Every incline or slope can be seen as a right triangle, allowing you to use trigonometry to find these components.
The vertical component represents the change in elevation, while the horizontal component represents the straight-line distance along the ground.

• **Vertical Components**: When ascending or descending, this is the height gain or loss, calculated as the sine of the angle times the length of the slope.
• **Horizontal Components**: This is the horizontal distance traveled, found by multiplying the cosine of the angle with the length of the slope.

In the given problem, this breakdown allows you to understand how each segment of the ride contributes to the overall path. While the total distance may seem like a straightforward addition of segment lengths, the rise in elevation is not directly additive, as it depends on the angle of each part. Understanding these components helps visualize and analyze the journey more effectively.