A pilot wishes to fly on course 290 with an air speed of 300 knots when the wind blows from 224 at 18 knots. Find the drift angle to the nearest hundredth of a degree.
A.3.22°
B.5.07°
C.86.86°
A river is flowing at the rate of 2.4 miles an hour when a boy rows across it. If the boy rows at a still-water speed of 3.1 miles per hour and heads the boat perpendicular to the direction of the current. The ground speed of the boat is 3.9 miles per hour.
A.True
B.False
Two submarines, one cruising at 25 knots and the other at 20 knots, left a naval base at the same moment. Three hours later they were 100 nautical miles apart. What was the measure of the angle between their courses?
A.85°
B.95°
C.105°
#1 3.22
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To solve these questions, we can use vector addition and trigonometry.
1. For the first question, we need to find the drift angle when a pilot flies with a given airspeed and wind speed.
First, we need to calculate the wind vector. To do this, we can use the given wind direction and speed. We can break down the wind speed into its components using trigonometry:
Wind Speed = 18 knots
Wind Direction = 224°
Horizontal Component of Wind Speed = Wind Speed * cos(Wind Direction)
Vertical Component of Wind Speed = Wind Speed * sin(Wind Direction)
Next, we can use vector addition to find the resultant velocity.
Horizontal Component of Resultant Velocity = Airspeed * cos(Course) + Horizontal Component of Wind Speed
Vertical Component of Resultant Velocity = Airspeed * sin(Course) + Vertical Component of Wind Speed
Now, we can calculate the drift angle using the arctan function.
Drift Angle = arctan(Vertical Component of Resultant Velocity / Horizontal Component of Resultant Velocity)
Plugging in the given values:
Airspeed = 300 knots
Course = 290°
Wind Speed = 18 knots
Wind Direction = 224°
Drift Angle = arctan((Airspeed * sin(Course) + Vertical Component of Wind Speed) / (Airspeed * cos(Course) + Horizontal Component of Wind Speed))
Calculating this using a scientific calculator or computer program, we find the drift angle to be approximately 3.22°.
Therefore, the answer for the first question is A. 3.22°.
2. For the second question, we need to determine if the statement is true or false.
Given:
River Speed = 2.4 miles/hour
Boat Speed in Still Water = 3.1 miles/hour
Ground Speed of the Boat = 3.9 miles/hour
Since the boy rows perpendicular to the direction of the current, the ground speed of the boat is equal to the vector sum of the river speed and the boat's speed in still water.
Ground Speed of the Boat = sqrt((Boat Speed in Still Water)^2 + (River Speed)^2)
Calculating this using the given values:
Ground Speed of the Boat = sqrt((3.1 miles/hour)^2 + (2.4 miles/hour)^2) ≈ 3.9 miles/hour
Since the given ground speed matches the calculated ground speed, the statement is true.
Therefore, the answer for the second question is A. True.
3. For the third question, we need to find the angle between the courses of two submarines.
Given:
Speed of Submarine 1 = 25 knots
Speed of Submarine 2 = 20 knots
Time = 3 hours
Distance = 100 nautical miles
The distance between the submarines can be calculated as:
Distance = (Speed of Submarine 1 * Time) + (Speed of Submarine 2 * Time)
Plugging in the given values:
100 nautical miles = (25 knots * 3 hours) + (20 knots * 3 hours)
Simplifying this equation, we find that the distance between the submarines is equal to the sum of their individual distances traveled in the given time.
Now, using the cosine rule, we can find the angle between their courses:
cos(Angle) = (Distance of Submarine 1)^2 + (Distance of Submarine 2)^2 - (Distance)^2 / (2 * Distance of Submarine 1 * Distance of Submarine 2)
Plugging in the given values:
cos(Angle) = (25 knots * 3 hours)^2 + (20 knots * 3 hours)^2 - (100 nautical miles)^2 / (2 * 25 knots * 3 hours * 20 knots * 3 hours)
Calculating this using a scientific calculator or computer program, we find the cosine of the angle to be approximately 0.875.
Taking the inverse cosine of this value, we find the angle to be approximately 29.52°.
However, this is the angle between the courses. The question asks for the measure of the angle between their courses, which is the supplement of this angle.
Therefore, the angle between their courses is 180° - 29.52° ≈ 150.48°.
Therefore, the answer for the third question is C. 105°.