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Chapter 3: Problem 32

To get to a concert in time, a harpsichordist has to drive \(122 \mathrm{mi}\) in \(2.00 \mathrm{h} .\) (a) If he drove at an average speed of \(55.0 \mathrm{mi} / \mathrm{h}\) in a due west direction for the first $1.20 \mathrm{h}\( what must be his average speed if he is heading \)30.0^{\circ}$ south of west for the remaining 48.0 min? (b) What is his average velocity for the entire trip?

Short Answer

Expert verified
a) Required average speed for the remaining part of the journey: ___ mi/h. b) Average velocity for the entire trip: ___ mi/h.

Step by step solution

01

Calculate the distance traveled in the first 1.20 h

The formula for distance is: Distance = Speed × Time We've been given the average speed during the first 1.20 h (55 mi/h) and the time interval (1.20 h). The distance traveled in the first part of the journey is: Distance_traveled_part1 = 55 mi/h × 1.20 h = 66 mi.
02

Find the remaining distance to be covered

Now, we'll subtract the distance traveled in the first 1.20 h (66 mi) from the total distance (122 mi) to find out the remaining distance to be covered: Remaining_distance = Total_distance - Distance_traveled_part1 Remaining_distance = 122 mi - 66 mi = 56 mi
03

Calculate the time for the remaining part of the journey

The remaining time in the journey is given as 48 minutes. To make it easier to compare with the time in hours, we can convert the remaining time to hours: Remaining_time = 48 min ÷ 60 min/h = 0.8 h
04

Find the required average speed for the remaining part

Now we can use the remaining distance and the remaining time to calculate the required average speed for the remaining part. Required_average_speed = Remaining_distance ÷ Remaining_time Required_average_speed = 56 mi ÷ 0.8 h = 70 mi/h Answer for (a): The required average speed for the remaining part of the journey is 70 mi/h.
05

Find the horizontal and vertical distances traveled

1. For the first part of the journey, the harpsichordist traveled only in the westward direction (horizontal), so the horizontal_distance_part1 = 66 mi and vertical_distance_part1 = 0 mi. 2. For the remaining part, we need to determine the horizontal and vertical components of the distance. We will use the remaining distance (56 mi) and the given direction (30 degrees south of west) to do so: Horizontal_distance_part2 = 56 mi × cos(30°) = 56 mi × 0.866 = 48.5 mi Vertical_distance_part2 = 56 mi × sin(30°) = 56 mi × 0.5 = 28 mi (southwards)
06

Calculate the total horizontal and vertical distances traveled

Now let's find the total horizontal and vertical distances traveled in the entire journey: Total_horizontal_distance = Horizontal_distance_part1 + Horizontal_distance_part2 Total_horizontal_distance = 66 mi + 48.5 mi = 114.5 mi Total_vertical_distance = Vertical_distance_part1 + Vertical_distance_part2 Total_vertical_distance = 0 mi + 28 mi = 28 mi
07

Compute the average velocity for the entire trip

Now, we can determine the resultant distance of the entire journey using the Pythagorean theorem: Resultant_distance = sqrt(Total_horizontal_distance² + Total_vertical_distance²) Resultant_distance = sqrt(114.5 mi² + 28 mi²) = sqrt(13156.25 mi²) ≈ 114.7 mi We also know that the total time for the journey is 2 h. We can now compute the average velocity: Average_velocity = Resultant_distance ÷ Total_time Average_velocity = 114.7 mi ÷ 2 h = 57.35 mi/h Answer for (b): The average velocity for the entire trip is approximately 57.35 mi/h.

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