Vector Question

Question

To make the penalty kick in a soccer game, Mia knows that the ball must travel (r) at 60 mph to the opposite goal post, which would be on a 70° angle with the field lines. However, she must account for a wind (w) coming from her right side at about 10 mph. Determine the angle and speed Mia should  use to kick the ball (k) to get the desired result.

Answer 1

Step 1: Set up coordinate system

1. x-axis: along field line toward goal (0°)
2. y-axis: perpendicular to field line (to her left at 90°)
3. Wind from right pushes ball left

Step 2: Write vectors in component form

Resultant r:

1. r_x = 60 cos(70°) = 60 × 0.342 = 20.52 mph
2. r_y = 60 sin(70°) = 60 × 0.9397 = 56.38 mph
3. r = (20.52, 56.38)

Wind w:

1. Blowing from right to left at 10 mph
2. w = (0, 10)

Step 3: Calculate kick vector

k = r - w = (20.52, 56.38) - (0, 10) = (20.52, 46.38)

Step 4: Find magnitude and angle

Speed: |k| = √(20.52² + 46.38²) = √(421.07 + 2151.11) = √2572.18 = 50.7 mph

Angle: θ = arctan(46.38/20.52) = arctan(2.26) = 66.1°