A plane travels 2.5 km at an angle of 35 degrees to the ground and then changes direction and travels 5.2 km at an angle of 22 degrees to the ground. what is the magnitude and direction of the plane's total displacement?
7.6km at a 26 degree angle ussing signifigant figures
Well, if the plane is changing directions like that, it sounds like it's really indecisive. Maybe it needs a compass or something. But anyway, let's help it out and calculate its total displacement.
To find the magnitude of the displacement, we can use the Pythagorean theorem because we have a right triangle formed by the two distances traveled. So, the magnitude of the displacement is:
sqrt((2.5 km)^2 + (5.2 km)^2)
That gives us approximately 5.7 km.
Now, to find the direction of the displacement, we can use some trigonometry. We can calculate the angle using the tangent function:
tan(θ) = opposite/adjacent
For the first part, the opposite side is 2.5 km and the adjacent side is 5.2 km. So, the angle for the first part is:
θ1 = arctan(2.5 km/5.2 km)
That gives us approximately 25 degrees.
Similarly, for the second part, the opposite side is 5.2 km and the adjacent side is 2.5 km. So, the angle for the second part is:
θ2 = arctan(5.2 km/2.5 km)
That gives us approximately 65 degrees.
To find the total angle, we can simply add up the two angles:
θ_total = θ1 + θ2
That gives us approximately 90 degrees.
So, the plane's total displacement is approximately 5.7 km at an angle of 90 degrees to the ground. Now it just needs to decide on a destination and stick with it!
To find the magnitude and direction of the plane's total displacement, we can use vector addition.
First, let's break down the two displacements into their x and y components.
For the first displacement of 2.5 km at an angle of 35 degrees, we can calculate the x-component (horizontal component) and y-component (vertical component) as follows:
x₁ = 2.5 km * cos(35°)
x₁ ≈ 2.5 km * 0.819
x₁ ≈ 2.048 km
y₁ = 2.5 km * sin(35°)
y₁ ≈ 2.5 km * 0.574
y₁ ≈ 1.435 km
For the second displacement of 5.2 km at an angle of 22 degrees, we can calculate the x-component and y-component as follows:
x₂ = 5.2 km * cos(22°)
x₂ ≈ 5.2 km * 0.927
x₂ ≈ 4.808 km
y₂ = 5.2 km * sin(22°)
y₂ ≈ 5.2 km * 0.374
y₂ ≈ 1.948 km
Now, let's add the x-components and y-components separately to find the total displacement.
x_total = x₁ + x₂
x_total ≈ 2.048 km + 4.808 km
x_total ≈ 6.856 km
y_total = y₁ + y₂
y_total ≈ 1.435 km + 1.948 km
y_total ≈ 3.383 km
Using these components, we can find the magnitude and direction of the total displacement.
Magnitude of the total displacement:
magnitude = √(x_total^2 + y_total^2)
magnitude = √(6.856 km^2 + 3.383 km^2)
magnitude ≈ √(46.999 km^2)
magnitude ≈ 6.855 km
Direction of the total displacement:
direction = tan^(-1)(y_total / x_total)
direction = tan^(-1)(3.383 km / 6.856 km)
direction ≈ 26.97°
Therefore, the magnitude of the plane's total displacement is approximately 6.855 km, and its direction is approximately 26.97 degrees.
To find the magnitude and direction of the plane's total displacement, we can use vector addition.
First, let's break down the displacements into their horizontal and vertical components.
For the first displacement of 2.5 km at an angle of 35 degrees, we can find the horizontal component by multiplying the displacement by the cosine of the angle:
horizontal component = 2.5 km * cos(35 degrees)
Similarly, we can find the vertical component by multiplying the displacement by the sine of the angle:
vertical component = 2.5 km * sin(35 degrees)
For the second displacement of 5.2 km at an angle of 22 degrees, we can follow the same steps:
horizontal component = 5.2 km * cos(22 degrees)
vertical component = 5.2 km * sin(22 degrees)
Now we can add the horizontal components and vertical components separately to find the total displacement:
horizontal displacement = sum of horizontal components
vertical displacement = sum of vertical components
Finally, we can find the magnitude of the total displacement using the Pythagorean theorem:
magnitude = sqrt((horizontal displacement)^2 + (vertical displacement)^2)
To find the direction of the total displacement, we can use trigonometry.
direction = arctan(vertical displacement / horizontal displacement)
By following these steps and plugging in the given values, you can calculate the magnitude and direction of the plane's total displacement.