Word Problems pages 18-19
A sailboat heads north at 3 m/s for 1 hour and then tracks back to the southeast (at 45 degrees to north) at 2 m/s for 45 minutes.
a. How far has the boat sailed?
b. How far is it from its starting location?
The textbook that included the word problem is: Basic Biomechanics Sixth Edition by Susan J. Hall;
ISBN#978-0-07-337644-8
The course name: Biomechanics
figure the distances in both directions.
distance=veloicyt*time
Add the distances. I would do this by taking the second direction as a composite of two vectors, one N, and one E.
Add the N component to the original course. You can now add the two ninety degree vectors.
To solve this word problem, we can break it down into two parts: the boat's northward journey and its southeastward journey.
a. To find how far the boat has sailed, we need to calculate the distance traveled during each part of the journey and then add them together.
During the northward journey, the boat travels at a speed of 3 m/s for 1 hour. Therefore, the distance traveled can be calculated using the formula:
Distance = Speed × Time
Distance = 3 m/s × 1 hour = 3 meters.
During the southeastward journey, the boat travels at a speed of 2 m/s for 45 minutes. However, we need to convert the time from minutes to hours by dividing it by 60.
Time = 45 minutes ÷ 60 = 0.75 hours.
Now we can calculate the distance traveled during the southeastward journey:
Distance = 2 m/s × 0.75 hours = 1.5 meters.
Finally, we can add the distances from both journeys to find the total distance the boat has sailed:
Total Distance = Distance of Northward Journey + Distance of Southeastward Journey
Total Distance = 3 meters + 1.5 meters = 4.5 meters.
Therefore, the boat has sailed a total distance of 4.5 meters.
b. To find how far the boat is from its starting location, we can use the concept of vectors. By breaking down the southeastward journey into its eastward and southward components, we can use the Pythagorean theorem to find the resultant displacement from the starting location.
During the southeastward journey, the boat moves at a 45-degree angle to the north. This angle can be decomposed into its eastward and southward components using trigonometry.
The eastward component:
e = 2 m/s × cos(45°)
e ≈ 2 m/s × 0.707 ≈ 1.414 m/s
The southward component:
s = 2 m/s × sin(45°)
s ≈ 2 m/s × 0.707 ≈ 1.414 m/s
Next, we can find the total eastward and southward distances traveled during the southeastward journey:
Eastward Distance = Eastward Speed × Time
Eastward Distance = 1.414 m/s × 0.75 hours ≈ 1.0615 meters
Southward Distance = Southward Speed × Time
Southward Distance = 1.414 m/s × 0.75 hours ≈ 1.0615 meters
Now, using the Pythagorean theorem, we can find the resultant displacement:
Resultant Displacement = √((Eastward Distance)^2 + (Southward Distance)^2)
Resultant Displacement = √(1.0615^2 + 1.0615^2)
Resultant Displacement ≈ √(1.1266 + 1.1266)
Resultant Displacement ≈ √2.2532
Resultant Displacement ≈ 1.5016 meters
Therefore, the boat is approximately 1.5016 meters from its starting location.
Please note that these calculations are based on the information given in the word problem, and the textbook provided is for reference purposes.