Two soccer players, Mia and Alice, are running as Alice passes the ball to Mia. Mia is running due north with a speed of 6.00 m/s. The velocity of the ball relative to Mia is 5.00 m/s in a direction 30.0 east of south. What are the magnitude and direction of the velocity of the ball relative to the ground?

v_b = 10.628 m / s (13.605 degrees east of south)

Explanation:

Given:

- Velocity of mia v_m = + 6 j m/s

- Velocity of ball wrt mia v_bm = 5.0 m/s 30 degree due east of south

Find:

What are the magnitude and direction of the velocity of the ball relative to the ground? v_b

Solution:

- The relation of velocity in two different frame is given:

v_b - v_m = v_bm

- Components along the direction of v_b, m:

v_b*cos (Q) - v_m*cos (30) = 5

v_b*cos (Q) = 5 + 6 sqrt (3) / 2

v_b*cos (Q) = 5 + 3sqrt (3)

- Components orthogonal the direction of v_b, m:

-v_m*sin (30) = v_b*sin (Q)

-6*0.5 = v_b*sin (Q)

-3 = v_b*sin (Q)

- Divide two equations:

tan (Q) = - 3 / 5 + 3sqrt (3)

Q = arctan ( - 3 / 5 + 3sqrt (3)

Q = - 16.395 degrees

v_b = - 3 / sin (-16.395)

v_b = 10.63 m/s