a plane travels 2.5 km at an angle of 35degrees to the ground then changes direction and travels 5.2 km at an angle of 22degrees to the ground. What is the magnitude and direction of the plane's displacement?
I will be happy to critique your work or thinking.
I have no clue how to do this. I can draw two triangles but than im lost
To find the magnitude and direction of the plane's displacement, we can represent the given information as vectors and use vector addition. Let's break down the problem step by step:
Step 1: Convert the given distances into their horizontal and vertical components. We can do this by using trigonometric functions.
For the first leg of the journey:
- Distance = 2.5 km
- Angle = 35 degrees
The horizontal component can be found by using the cosine function:
Horizontal component = 2.5 km * cos(35 degrees)
The vertical component can be found by using the sine function:
Vertical component = 2.5 km * sin(35 degrees)
Similarly, for the second leg of the journey:
- Distance = 5.2 km
- Angle = 22 degrees
Horizontal component = 5.2 km * cos(22 degrees)
Vertical component = 5.2 km * sin(22 degrees)
Step 2: Add the horizontal components and the vertical components separately to find the resultant displacement in the horizontal and vertical directions.
Resultant horizontal component = (Horizontal component of first leg) + (Horizontal component of second leg)
Resultant vertical component = (Vertical component of first leg) + (Vertical component of second leg)
Step 3: Calculate the magnitude of the displacement using the Pythagorean theorem.
Magnitude of displacement = sqrt((Resultant horizontal component)^2 + (Resultant vertical component)^2)
Step 4: Calculate the direction of the displacement using the inverse tangent function.
Direction (in degrees) = arctan(Resultant vertical component / Resultant horizontal component)
Now, let's plug in the values and calculate the displacement:
Horizontal component of first leg = 2.5 km * cos(35 degrees) ≈ 2.043 km
Vertical component of first leg = 2.5 km * sin(35 degrees) ≈ 1.459 km
Horizontal component of second leg = 5.2 km * cos(22 degrees) ≈ 4.657 km
Vertical component of second leg = 5.2 km * sin(22 degrees) ≈ 1.919 km
Resultant horizontal component = 2.043 km + 4.657 km = 6.7 km (rounded to one decimal place)
Resultant vertical component = 1.459 km + 1.919 km = 3.378 km (rounded to one decimal place)
Magnitude of displacement = sqrt((6.7 km)^2 + (3.378 km)^2) ≈ 7.5 km (rounded to one decimal place)
Direction = arctan(3.378 km / 6.7 km) ≈ 26.9 degrees (rounded to one decimal place)
Therefore, the magnitude of the plane's displacement is approximately 7.5 km, and the direction is approximately 26.9 degrees.