A plane is steering east at a speed of 240km/h. What is the ground speed of the plane if the wind is from the northwest at 65km/h. What is the plane's actual direction?
speed relative to ground: 240E+65*cos45E+65*sin45S
combine like directions, then directionactual= arctan(S/E) south of East.
resultant = 240(1,0) + 65(cos315, sin315)
= (240,0) + (45.962, -45.962)
= (285.962, -45.962)
|resultant| = √(285.962^2 + (-45.962^2 )
= 289.63 km/h
angle: tanØ = -45.962 ÷ 285.962 = ....
Ø = 350.9°°
or S 80.9*° E
or by cosine law:
R^2 = 240^2 + 65^2 - 2(240)(65)cos135°
= 83886.73..
R = 289.63
now use the Sine Law to find the angle
To find the ground speed of the plane, we can use vector addition.
Step 1: Draw a diagram depicting the velocity of the plane and the velocity of the wind.
Step 2: Resolve the velocity of the wind into its northward and eastward components. The northwest wind can be resolved into a northward component and an eastward component.
Given the northwest wind speed is 65 km/h, we can split it into two perpendicular components. Using trigonometry, we can find:
Northward component = 65 km/h * sin(45°)
Eastward component = 65 km/h * cos(45°)
Step 3: Add the eastward component of the wind to the eastward velocity of the plane.
240 km/h (plane's eastward velocity) + Eastward component (wind's eastward component) = Ground speed
Step 4: Add the northward component of the wind to the northward velocity of the plane.
0 km/h (plane's northward velocity) + Northward component (wind's northward component) = Ground speed
Step 5: Calculate the magnitude and direction of the resulting velocity vector.
To find the magnitude (ground speed), use the Pythagorean theorem:
Ground speed = sqrt((eastward component of velocity)^2 + (northward component of velocity)^2)
To find the angle, use the inverse tangent:
Plane's actual direction = arctan((northward component of velocity) / (eastward component of velocity))
Calculating the values:
Eastward component = 65 km/h * cos(45°) ≈ 45.9 km/h
Northward component = 65 km/h * sin(45°) ≈ 45.9 km/h
Ground speed = sqrt((240 km/h + 45.9 km/h)^2 + (0 km/h + 45.9 km/h)^2) ≈ sqrt(291.9 km/h^2 + 45.9 km/h^2) ≈ sqrt(35330.42 km^2/h^2) ≈ 187.9 km/h
Plane's actual direction = arctan((0 km/h + 45.9 km/h) / (240 km/h + 45.9 km/h)) ≈ arctan(45.9 km/h / 285.9 km/h) ≈ 9.10°
Therefore, the ground speed of the plane is approximately 187.9 km/h, and the plane's actual direction is approximately 9.10 degrees east of due north.
To find the ground speed of the plane, we need to consider the effect of the wind's speed and direction on the plane's motion. We can break down the plane's motion into two components: its airspeed and its heading.
1. Determine the airspeed: The airspeed of the plane is given as 240 km/h. This is the speed of the plane relative to the air, unaffected by wind.
2. Analyze the wind's effect: The wind is blowing from the northwest at a speed of 65 km/h. Since it is from the northwest, we need to consider its components in the eastward and northward directions.
- Eastward component of the wind: To determine the eastward component, we can use trigonometry. Since the wind is from the northwest, it forms a 45-degree angle with the eastward direction. So, the eastward component of the wind is 65 km/h multiplied by cos(45°).
- Northward component of the wind: Similarly, the northward component of the wind is 65 km/h multiplied by sin(45°).
3. Determine the plane's actual direction: The plane's actual direction combines its heading and the direction of the wind. To find the actual direction, we can add the vectors representing the plane's heading and the wind's effect.
- Heading: The plane is steering east, so its heading is directly to the east, which has a compass bearing of 90°.
- Wind's effect: The wind's eastward and northward components combine with the plane's heading.
4. Calculate the ground speed: The ground speed is the magnitude of the resultant vector obtained from adding the heading and the wind's effect.
Now let's calculate the values:
Eastward component of the wind = 65 km/h * cos(45°) ≈ 45.91 km/h (rounded to two decimal places)
Northward component of the wind = 65 km/h * sin(45°) ≈ 45.91 km/h (rounded to two decimal places)
Adding the heading (east) with the wind's effect gives us a vector of (240 km/h + 45.91 km/h) in the eastward direction and (45.91 km/h) in the northward direction.
Using the Pythagorean theorem, the magnitude of the resultant vector (ground speed) is:
Ground speed = √[(240 km/h + 45.91 km/h)^2 + (45.91 km/h)^2] ≈ 251.11 km/h (rounded to two decimal places)
The plane's actual direction can be calculated using trigonometry:
Actual direction = arctan([eastward component of the resultant vector] / [northward component of the resultant vector])
Actual direction = arctan(286.91 km/h / 45.91 km/h) ≈ 82.1° (rounded to one decimal place)
Therefore, the ground speed of the plane is approximately 251.11 km/h, and the plane's actual direction is approximately 82.1°.