A plane is flying with an airspeed of 150 miles per hour and heading 155°. The wind currents are running at 25 miles per hour at 160° clockwise from due north. Use vectors to find the true course and ground speed of the plane.
To find the true course and ground speed of the plane, we can use vector addition.
Step 1: Convert the given information into vector form.
The airspeed of the plane can be represented as a vector, which we'll call A. A has a magnitude of 150 mph and is directed at an angle of 155°.
The wind current can also be represented as a vector, which we'll call W. W has a magnitude of 25 mph and is directed at an angle of 160° clockwise from due north.
Step 2: Convert the vectors into component form.
To add the vectors, we need to convert them into their horizontal (x) and vertical (y) components. We can use trigonometry for this.
For vector A:
Ax = magnitude of A * cos(angle of A)
Ay = magnitude of A * sin(angle of A)
Ax = 150 * cos(155°)
Ax ≈ -109.11 mph
Ay = 150 * sin(155°)
Ay ≈ 41.04 mph
For vector W:
Wx = magnitude of W * cos(angle of W)
Wy = magnitude of W * sin(angle of W)
Wx = 25 * cos(160°)
Wx ≈ 11.76 mph
Wy = 25 * sin(160°)
Wy ≈ -4.25 mph
Step 3: Calculate the total vector.
To find the total vector, we add the x-components and the y-components separately.
Total x-component = Ax + Wx
Total x-component ≈ -109.11 mph + 11.76 mph
Total x-component ≈ -97.35 mph
Total y-component = Ay + Wy
Total y-component ≈ 41.04 mph - 4.25 mph
Total y-component ≈ 36.79 mph
Step 4: Find the magnitude and angle of the total vector.
The magnitude of the total vector can be found using the Pythagorean theorem.
Magnitude = sqrt((Total x-component)^2 + (Total y-component)^2)
Magnitude ≈ sqrt((-97.35 mph)^2 + (36.79 mph)^2)
Magnitude ≈ sqrt(9471.92 + 1350.44)
Magnitude ≈ sqrt(10822.36)
Magnitude ≈ 103.97 mph (rounded to two decimal places)
The angle of the total vector can be found using the inverse tangent function.
Angle = atan(Total y-component / Total x-component)
Angle ≈ atan(36.79 mph / -97.35 mph)
Angle ≈ atan(-0.3776)
Angle ≈ -20.18° (rounded to two decimal places)
Step 5: Interpret the results.
The true course of the plane is the direction of the total vector. In this case, it is approximately -20.18°.
The ground speed of the plane is the magnitude of the total vector. In this case, it is approximately 103.97 mph.
To find the true course and ground speed of the plane, we can break down the velocities into their vector components and then add them up.
Let's start by finding the x and y components of the airspeed and the wind currents.
Airspeed:
The airspeed is given as 150 miles per hour at a heading of 155°. To find the x and y components, we can use the following trigonometric relationships:
V_airspeed_x = V_airspeed * cos(heading)
V_airspeed_y = V_airspeed * sin(heading)
where V_airspeed is the airspeed and heading is the angle in degrees.
Using these formulas, we can calculate the components of the airspeed:
V_airspeed_x = 150 * cos(155°)
V_airspeed_y = 150 * sin(155°)
Wind Currents:
The wind currents are given as 25 miles per hour at an angle of 160° clockwise from due north. We can use similar formulas to find the x and y components:
V_wind_x = V_wind * cos(wind_angle)
V_wind_y = V_wind * sin(wind_angle)
where V_wind is the wind speed and wind_angle is the angle in degrees.
Calculating the components of the wind currents:
V_wind_x = 25 * cos(160°)
V_wind_y = 25 * sin(160°)
Now, we can add up the x and y components of the airspeed and wind currents to find the total x and y components of the resulting velocity, which will give us the ground speed and direction (true course) of the plane.
V_ground_x = V_airspeed_x + V_wind_x
V_ground_y = V_airspeed_y + V_wind_y
Finally, we can calculate the magnitude and direction of the resulting velocity vector:
Ground speed = √(V_ground_x^2 + V_ground_y^2)
True course = arctan(V_ground_y / V_ground_x)
Plugging in the values we calculated earlier will give us the true course and ground speed of the plane.
150 at 155° = 63.4i -135.9j
25 at 160° = 8.6i - 23.5j
sum = 72.0i - 159.4j = 175.0 at 155.7°