In a World Cup soccer match, Juan is running due north toward the goal with a speed of 7.60m/s relative to the ground. A teammate passes the ball to him. The ball has a speed of 13.9m/s and is moving in a direction of 36.0❝ east of north, relative to the ground.

Part A

What is the magnitude of the ball's velocity relative to Juan?

Part B

What is the direction of the ball's velocity relative to Juan?

a) The magnitude of the ball's velocity relative to Juan = 8.95 m/s

b) The direction of the ball's velocity relative to Juan is 24.1° from the + x-axis or N 65.9° E, that is, 65.9° east of North.

Explanation:

Relative velocity of a body X with respect to another body Y (Vₓᵧ) = velocity of Body X (Vₓ) - velocity of Body Y (Vᵧ)

Velocity of Juan in vector form = 7.6j m/s

Velocity of the ball in vector form = 13.9 [ (sin 36°) î + (cos 36°) ] (it has this form because the angle is given with respect to the y-axis, that is, N 36° E)

Velocity of the ball in vector form = (8.17î + 11.25j) m/s

a) The ball's velocity relative to Juan = (velocity of the ball) - (velocity of Juan) = (8.17î + 11.25j) - (7.6j) = (8.17î + 3.65j) m/s

Magnitude = √[ (8.17) ² + (3.65) ²] = 8.95 m/s

b) direction of the ball relative to Juan = direction of the vector obtained in (a)

θ = Tan⁻¹ (3.65/8.17)

θ = 24.1°