A wind is blowing directly from east to west. The pilot of a small plane finds that if he points the nose of the plane 30.0 ° north of east, his velocity with respect to the ground is in the direction 55.1 ° north of east. The speed of the plane with respect to the air is 126 m/s. Taking North to be the y-direction and East to be the x-direction, what is the y-component of plane's velocity with respect to the ground?
To find the y-component of the plane's velocity with respect to the ground, we need to break down the components of the plane's velocity vector using trigonometry.
Let's define the x-direction as east and the y-direction as north. We have two pieces of information: the angle the pilot points the nose of the plane with respect to the x-axis (30.0° north of east) and the angle the velocity vector makes with the x-axis (55.1° north of east).
First, let's find the x-component of the plane's velocity with respect to the ground. Since the plane is flying directly east to west, the x-component of its velocity remains unaffected by the wind. Therefore, the x-component of the velocity with respect to the ground is the same as the x-component of the velocity with respect to the air:
Vx = Vairx = Vair * cos(theta)
where Vair is the speed of the plane with respect to the air (126 m/s) and theta is the angle the pilot points the nose of the plane (30.0°).
Vx = 126 m/s * cos(30.0°)
Vx ≈ 109.27 m/s (rounded to two decimal places)
Now, let's find the y-component of the plane's velocity with respect to the ground. Since the wind is blowing directly from east to west, we can consider it as an additional velocity vector that adds to the plane's velocity vector. The y-component of the plane's velocity with respect to the ground is the sum of the y-component of the velocity with respect to the air and the wind's y-component:
Vy = Vairy + Vwindy
We can find Vairy using the following equation:
Vairy = Vair * sin(theta)
where Vair is the speed of the plane with respect to the air (126 m/s) and theta is the angle the pilot points the nose of the plane (30.0°).
Vairy = 126 m/s * sin(30.0°)
Vairy ≈ 63.00 m/s (rounded to two decimal places)
To find Vwindy, we can use the fact that the angle the velocity vector makes with the x-axis is 55.1° north of east. The y-component of the velocity vector with respect to the ground is given by:
Vwindy = Vwind * sin(alpha)
where Vwind is the speed of the wind and alpha is the angle the velocity vector makes with the x-axis (55.1°). Since we know the direction of the wind is from east to west, Vwind is simply the opposite of Vair in the x-direction. Therefore, Vwind is equal to -Vx.
Vwindy = -Vx * sin(alpha)
Vwindy = -109.27 m/s * sin(55.1°)
Vwindy ≈ -92.98 m/s (rounded to two decimal places)
Finally, we can calculate the y-component of the plane's velocity with respect to the ground:
Vy = Vairy + Vwindy
Vy = 63.00 m/s + (-92.98 m/s)
Vy ≈ -29.98 m/s (rounded to two decimal places)
Therefore, the y-component of the plane's velocity with respect to the ground is approximately -29.98 m/s.