A plane takes 1.2 h to fly to a point that is 420 km directly west of its original position if the wind was blowing at a velocity of 65km/h [E 25degree N] what was the velocity of the plane relative to the wind?
plane: p = 420km/1.2hr = 350 @ W
wind: w = 65 @ E25N
p relative to w is just p-w so subtract the vectors in the usual way.
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Vp + Vw = 420/1.2 = 350km/h[180o].
Vp + 65[25o] = 350[18o],
Vp = 350[180] - 65[25o],
Vp = (350*cos180-65*cos25) + (350*sin180-65*sin25)I,
Vp = -409 - 27.5i = 410km/h[3.85o] S. of W.
To find the velocity of the plane relative to the wind, we need to break down the given information and use vector addition.
Let's first consider the velocity of the plane relative to the ground. We know that the plane flew 420 km directly west, and it took 1.2 hours to do so. Therefore, the velocity of the plane relative to the ground (without considering the wind) can be calculated by dividing the distance by the time:
Velocity (plane relative to ground) = Distance / Time
= 420 km / 1.2 h
= 350 km/h (west)
Now, we need to consider the effect of the wind. The wind is blowing at a velocity of 65 km/h [E 25° N]. To determine the effect of the wind on the plane's velocity, we can break the wind velocity into horizontal (east-west) and vertical (north-south) components.
Horizontal component of wind velocity = 65 km/h * cos(25°) [east]
≈ 58.660 km/h [east]
Vertical component of wind velocity = 65 km/h * sin(25°) [north]
≈ 27.747 km/h [north]
Now, we can determine the overall effect of the wind on the plane's velocity by subtracting the wind's eastward component from the plane's westward velocity:
Velocity (plane relative to the wind) = Velocity (plane relative to ground) - Horizontal component of wind velocity
= 350 km/h (west) - 58.660 km/h (east)
= 291.340 km/h (west)
Therefore, the velocity of the plane relative to the wind is approximately 291.340 km/h (west).