A speedboat heads west at 104 km/h for 24.0 min. It then travels at 64.0° south of west at 109.0 km/h for 13.0 min.
(a) What is the average speed for the trip?
What is the average velocity for the trip?
magnitude
direction
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total distance traveled = (24/60)(104) + (13/60)(109) = 65.216666..
total time = 37 minutes = 37/60 hrs
average speed = 65.21666../(37/60) = 105.8 km/h
for the average velocity, make a vector diagram
I have a triangle ABC with AB = 41.6 , BC = 23.1666..
and angle ABC = 116°
By the cosine law:
AC^2 = AB^2 + BC^2 - 2(AB)(BC)cos 116°
etc
(I will let you do the arithmetic)
The direction angle would be angle BAC
since you now have all 3 sides of the triangle plus an angle you can find angle BAC by the sine law.
To find the average speed for the trip, you need to calculate the total distance traveled and divide it by the total time taken.
For the first leg of the journey, the speedboat is heading west at 104 km/h for 24.0 minutes. To convert minutes to hours, divide by 60:
24.0 min ÷ 60 min/h = 0.4 h
The distance traveled in this leg can be calculated using the formula: distance = speed × time:
Distance (westward leg) = 104 km/h × 0.4 h = 41.6 km
For the second leg of the journey, the speedboat is traveling at 64.0° south of west at 109.0 km/h for 13.0 minutes. To convert minutes to hours:
13.0 min ÷ 60 min/h = 0.217 h
The distance traveled in this leg can be calculated using the formula: distance = speed × time:
Distance (southwest leg) = 109.0 km/h × 0.217 h = 23.653 km
Now, find the total distance traveled by adding the distances of both legs:
Total distance = Distance (westward leg) + Distance (southwest leg) = 41.6 km + 23.653 km = 65.253 km
Next, find the total time taken by adding the times of both legs:
Total time = 24.0 min + 13.0 min = 37.0 min = 37.0 min ÷ 60 min/h = 0.617 h
Finally, to find the average speed for the trip, divide the total distance by the total time:
Average speed = Total distance / Total time = 65.253 km / 0.617 h = 105.88 km/h
Now, to find the average velocity for the trip, you need to consider both the magnitude and the direction. The magnitude of the average velocity is the same as the average speed, which is 105.88 km/h.
To find the direction of the average velocity, we need to use vector addition. The westward leg has a velocity of 104 km/h to the west, and the southwest leg has a velocity of 109.0 km/h at a 64.0° angle south of west. We can use trigonometry to calculate the horizontal and vertical components of the southwest velocity:
Horizontal component = 109.0 km/h × cos(64.0°) = 49.751 km/h
Vertical component = 109.0 km/h × sin(64.0°) = 95.154 km/h
Now, we can add the horizontal components of the velocities of both legs:
Horizontal velocity = 104 km/h + 49.751 km/h = 153.751 km/h
For the vertical components, the westward leg has no vertical velocity, so it remains 0 km/h:
Vertical velocity = 0 km/h + 95.154 km/h = 95.154 km/h
Using the horizontal and vertical components, we can now use trigonometry to find the direction of the average velocity. The angle can be found using the formula:
Angle = arctan(vertical velocity / horizontal velocity)
Angle = arctan(95.154 km/h / 153.751 km/h) = arctan(0.6189) = 31.12°
Therefore, the average velocity for the trip has a magnitude of 105.88 km/h and a direction of 31.12° south of west.