A child is splaying with a toy aero plane. the aero plane is flying with velocity (6i + 2j)m/s. A breeze begin to blow with velocity 0.5j m/s affecting the motion of the aero plane.

a) find the magnitude of the resultant velocity of the plane
b) find the angle this resultant velocity makes with the unit vector i

(6,2) + (0,.5) = (6,2.5)

|6,2.5| = 6.5
tanθ = 2.5/6
θ = 22.6°

To find the magnitude of the resultant velocity of the plane, we need to find the vector sum of the initial velocity of the airplane and the velocity of the breeze.

Given:
Initial velocity of the airplane = (6i + 2j) m/s
Velocity of the breeze = 0.5j m/s

To find the vector sum, we add the x-components and y-components separately.

Adding the x-components:
Initial velocity in the x-direction = 6i m/s
Because the breeze has no effect on the x-component, the x-component of the resultant velocity remains the same.
Resultant velocity in the x-direction = 6i m/s

Adding the y-components:
Initial velocity in the y-direction = 2j m/s
Effect of the breeze in the y-direction = 0.5j m/s
Resultant velocity in the y-direction = 2j m/s + 0.5j m/s = 2.5j m/s

Now we have the resultant velocity in the x-direction and y-direction. To find the magnitude of the resultant velocity, we can use the Pythagorean theorem.

Magnitude of the resultant velocity (V) = sqrt((x-component)^2 + (y-component)^2)
V = sqrt((6)^2 + (2.5)^2) = sqrt(36 + 6.25) = sqrt(42.25) ≈ 6.5 m/s

a) The magnitude of the resultant velocity of the airplane is approximately 6.5 m/s.

b) To find the angle the resultant velocity makes with the unit vector i, we can use trigonometry. The angle can be found using the formula:

Angle (θ) = arctan(y-component / x-component)

Angle (θ) = arctan(2.5 / 6) ≈ 21.8°

b) The angle the resultant velocity makes with the unit vector i is approximately 21.8°.