A plane flies North 40degrees East at a constant speed of 600km/h. If the velocity of the wind is 80k/h blowing 20 degrees South of East, what is the velocity of the plane relative to the ground?
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Sum the x- and y-components of both vectors:
Vx = 600cos(40°) + 80cos(20°)
Vx ≈ 459.6267 + 75.1754
Vx ≈ 534.8021 km/r
Vy = 600sin(40°) - 80sin(20°) ....... minus because the wind is blowing SOUTH and east.
Vy ≈ 385.6726 - 27.3616
Vy ≈ 358.3110
V = √(Vx² + Vy²)
V ≈ √(534.8021² + 358.3110²)
V ≈ √414400
V ≈ 643.74 km/h
θ = tan^-1(Vy/Vx)
θ ≈ tan^-1(358.3110/534.8021)
θ ≈ tan^-1(0.66999)
θ ≈ 33.8°
The plane's speed relative to the ground is 643.74 km/h North 33.8° East
Vx = 600cos(40°) + 80cos(20°)
Vx ≈ 459.6267 + 75.1754
Vx ≈ 534.8021 km/r
Vy = 600sin(40°) - 80sin(20°) ....... minus because the wind is blowing SOUTH and east.
Vy ≈ 385.6726 - 27.3616
Vy ≈ 358.3110
V = √(Vx² + Vy²)
V ≈ √(534.8021² + 358.3110²)
V ≈ √414400
V ≈ 643.74 km/h
θ = tan^-1(Vy/Vx)
θ ≈ tan^-1(358.3110/534.8021)
θ ≈ tan^-1(0.66999)
θ ≈ 33.8°
The plane's speed relative to the ground is 643.74 km/h North 33.8° East
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You need to sketch a diagram, so you can see what you're doing. I'll describe how I'd do it, but be aware that you might have been taught a slightly different method.
I would draw myself an arrow to show the plane's velocity relative to the air. So, that's an arrow pointing North 40° East (I assume this means 40° east of north), which I label '600 km/h'. I draw a dotted vertical line from the base of my arrow to show where north is, and I mark in the 40° angle.
Now, at this point some people would draw a second arrow from the point of the first, showing the wind blowing 80 km/h 20° south of east. And you might find that helps you think about what you're doing, and it's true that you actually have enough information about angles and 'lengths of sides' that you could make a triangle and find the resultant force. But I like working with perpendicular components.
So, I'd break the 'velocity of the plane with respect to the air' down into north and east components. Look at your arrow, with the dotted line for north. You can draw a dotted line east from your north line, such that it ends at the tip of your arrow. Now you have a triangle, with a known hypotenuse and a known angle. You can find how 'long' (hence, the speed) the north and east components are. Use trig.
Now do the same thing with the velocity of the wind: draw an arrow, but with a dotted line for 'east' instead of for north. Find the south and east components.
Now it's simple arithmetic to find the resultant north and east components (remember that a south component should be subtracted from a north component). You can sketch one final triangle, with your new north and east components, and use pythagoras to find the hypotenuse (resultant speed), and trig to find the angle from north.
I would draw myself an arrow to show the plane's velocity relative to the air. So, that's an arrow pointing North 40° East (I assume this means 40° east of north), which I label '600 km/h'. I draw a dotted vertical line from the base of my arrow to show where north is, and I mark in the 40° angle.
Now, at this point some people would draw a second arrow from the point of the first, showing the wind blowing 80 km/h 20° south of east. And you might find that helps you think about what you're doing, and it's true that you actually have enough information about angles and 'lengths of sides' that you could make a triangle and find the resultant force. But I like working with perpendicular components.
So, I'd break the 'velocity of the plane with respect to the air' down into north and east components. Look at your arrow, with the dotted line for north. You can draw a dotted line east from your north line, such that it ends at the tip of your arrow. Now you have a triangle, with a known hypotenuse and a known angle. You can find how 'long' (hence, the speed) the north and east components are. Use trig.
Now do the same thing with the velocity of the wind: draw an arrow, but with a dotted line for 'east' instead of for north. Find the south and east components.
Now it's simple arithmetic to find the resultant north and east components (remember that a south component should be subtracted from a north component). You can sketch one final triangle, with your new north and east components, and use pythagoras to find the hypotenuse (resultant speed), and trig to find the angle from north.