Question

An engineer designs a roller coaster so that a car travels horizontally for 172 ft, then climbs 147 ft at an angle of 31.0°above the horizontal. It then moves 147 ft at an angle of 40.0° below the horizontal. If we take the initial horizontal motion of the car to be along the +x-axis, what is the car's displacement?

Answer 1

The car's displacement is found by adding the horizontal and vertical components of each motion segment using vector addition. The first segment is purely horizontal, while the next two segments require decomposition into horizontal and vertical components using trigonometry. The sum of these components gives the total displacement vector.

Step-by-step explanation:

To find the car's displacement, we need to consider each segment of its journey and combine them using vector addition. The first segment is purely initial horizontal motion along the +x-axis for 172 ft. The next segment is at an angle of 31.0° above the horizontal and has a length of 147 ft. We can decompose this into horizontal (x) and vertical (y) components using trigonometry:

Horizontal component (x) = 147 ft × cos(31.0°)

Vertical component (y) = 147 ft × sin(31.0°)

The final segment is at an angle of 40.0° below the horizontal and also has a length of 147 ft. Its components are:

Horizontal component (x) = 147 ft × cos(40.0°)

Vertical component (y) = -147 ft × sin(40.0°) (negative because it's below the horizontal)

The total displacement is the sum of these components. The horizontal displacements add, while the vertical components combine accounting for direction (up is positive, down is negative). The final displacement is a vector that can be represented with its x and y components or as a magnitude and direction from the original position.