An airplane has a velocity relative to the air of 181 m/s in a westerly direction. If the wind has a speed relative to the ground of 53 m/s directed to the north, what is the speed of the plane relative to the ground?
Vpg = Vpa + Vag
Vpa = -181m/s @ 180deg.
Vag = 53m/s @ 90deg.
V(hor) = -181cos180 + 53cos90,
V(hor) = 181 + 0 = 181m/s.
V(ver) = -181sin180 + 53sin90,
V(ver) = 0 + 53 = 53m/s.
V^2 = X^2 + Y^2,
V^2 = (181)^2 + (53)^2,
V^2 = 32761 + 2809 = 35570,
V=sqrt(35570) = 189m/s=Speed of plane relative to ground.
DEFINITIONS
Vpg = Speed of plane relative to ground.
Vpa = Speed of plane relative to air.
Vag = Speed of air relative to ground.
To find the speed of the plane relative to the ground, we need to combine the velocity of the plane relative to the air with the velocity of the wind relative to the ground.
Let's break down the problem step by step:
1. Start by drawing a diagram to visualize the situation. Label the direction of the plane relative to the air as west and the direction of the wind relative to the ground as north. This will help us apply vector addition later.
2. We are given that the velocity of the plane relative to the air is 181 m/s in the west direction. This means the plane is moving against the wind.
3. We are also given that the velocity of the wind relative to the ground is 53 m/s in the north direction.
4. To find the speed of the plane relative to the ground, we will use vector addition. Since the velocities are in different directions, we can't simply add or subtract their magnitudes.
5. Convert both velocities (plane relative to air and wind relative to ground) into vectors. The velocity of the plane relative to the air is represented as a vector with a magnitude of 181 m/s pointing west. The velocity of the wind relative to the ground is represented as a vector with a magnitude of 53 m/s pointing north.
6. Add the two vectors together by forming a parallelogram using the two vectors as sides. The diagonal of the parallelogram will represent the resultant vector, which is the velocity of the plane relative to the ground.
7. Use the Pythagorean theorem to find the magnitude of the resultant vector. The magnitude is the square root of the sum of the squares of the components. In this case, the horizontal component (west) is the same as the plane's velocity relative to the air (181 m/s), and the vertical component (north) is the velocity of the wind relative to the ground (53 m/s).
So, the speed of the plane relative to the ground is √(181^2 + 53^2) m/s.
8. Calculate the magnitude of the resultant vector: √(181^2 + 53^2) = √(32761 + 2809) = √35570 ≈ 188.7 m/s.
Therefore, the speed of the plane relative to the ground is approximately 188.7 m/s.