After flying for 15 min in a wind blowing 42 km/h at an angle of 34° south of east, an airplane pilot is over a town that is 40 km due north of the starting point. What is the speed of the airplane relative to the air?
To find the speed of the airplane relative to the air, we need to consider the vector components of the velocity.
Let's break down the velocity into its north and east components:
Velocity in the north direction (Vnorth) = 40 km due north (since the town is directly north of the starting point)
Velocity in the east direction (Veast) = distance / time = (15 min)*(42 km/h) = 10.5 km (since velocity = distance / time)
Now we can find the resultant velocity using these components. The resultant velocity (Vresultant) is the vector sum of Vnorth and Veast.
Using the Pythagorean theorem, we can calculate the magnitude of the resultant velocity:
Vresultant = √(Vnorth^2 + Veast^2)
Vresultant = √((40 km)^2 + (10.5 km)^2)
Vresultant ≈ √(1600 km^2 + 110.25 km^2)
Vresultant ≈ √(1710.25 km^2)
Vresultant ≈ 41.4 km (rounded to one decimal place)
Hence, the speed of the airplane relative to the air is approximately 41.4 km/h.
To find the speed of the airplane relative to the air, we can use vector addition.
First, let's break down the given information:
- The wind is blowing at a velocity of 42 km/h at an angle of 34° south of east.
- The airplane has flown for 15 minutes, which is equivalent to 15/60 = 0.25 hours.
- The airplane is directly north of the starting point, forming a right triangle with the wind vector.
Now, let's calculate the distance the airplane has traveled relative to the wind. We know that speed = distance/time. In this case, the time is 0.25 hours.
Distance = speed × time
The distance traveled by the airplane can be represented as:
distance = speed of airplane relative to wind × time
Now, we need to find the horizontal and vertical components of the wind velocity. To do this, we can use trigonometry.
The horizontal component of the wind velocity is given by:
horizontal component = wind speed × cos(angle)
Substituting the given values:
horizontal component = 42 km/h × cos(34°)
Calculating this:
horizontal component ≈ 42 km/h × 0.829 (rounded to 3 decimal places)
horizontal component ≈ 34.82 km/h
The vertical component of the wind velocity is given by:
vertical component = wind speed × sin(angle)
Substituting the given values:
vertical component = 42 km/h × sin(34°)
Calculating this:
vertical component ≈ 42 km/h × 0.556 (rounded to 3 decimal places)
vertical component ≈ 23.35 km/h
Now, we have the horizontal and vertical components of the wind velocity. Since the airplane is flying north, the horizontal component of the airplane's velocity relative to the wind is equal to the horizontal component of the wind velocity.
Therefore, the horizontal component of the airplane's velocity relative to the air is 34.82 km/h.
Since the airplane is flying directly north, the vertical component of the airplane's velocity relative to the wind is equal to the speed of the airplane relative to the air.
Therefore, the vertical component of the airplane's velocity relative to the air is 23.35 km/h.
Now, we can find the speed of the airplane relative to the air using the Pythagorean theorem.
speed of airplane relative to air = √(horizontal component² + vertical component²)
Calculating this:
speed of airplane relative to air = √(34.82 km/h² + 23.35 km/h²)
speed of airplane relative to air ≈ √(1213.9524 km²/h²)
speed of airplane relative to air ≈ 34.81 km/h
Therefore, the speed of the airplane relative to the air is approximately 34.81 km/h.