Two soccer players on a breakaway start from midfield and run straight down the field with a separation of d=5 m and with the same velocity v=7 m/s. The last defensive player of the other team approaches the player who has the ball, so that person decides to pass the ball with velocity u=10 m/s to their teammate. What angle in degrees with respect to their trajectory should they pass the ball at so it reaches their teammate?

Question

Two soccer players on a breakaway start from midfield and run straight down the field with a separation of d=5 m and with the same velocity v=7 m/s. The last defensive player of the other team approaches the player who has the ball, so that person decides to pass the ball with velocity u=10 m/s to their teammate. What angle in degrees with respect to their trajectory should they pass the ball at so it reaches their teammate?

Answer 1

This problem has been posted on Brilliant as a current problem in this weeks' Geometry problem set.

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- Calvin Lin
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Answer 2

Answer 0.0001.

Answer 3

To find the angle at which the soccer player should pass the ball, we can use the concept of relative velocity. The relative velocity is the velocity of an object in relation to another object.

In this case, we have two velocities: the velocity of the ball before the pass (u) and the velocity of the players moving down the field (v).

To find the angle, we can break down the velocities into their horizontal and vertical components. Let's assume the x-axis is aligned with the direction of the players' trajectory and the y-axis is perpendicular to it.

1. Find the horizontal components of both velocities:
- The horizontal component of the ball's velocity (u) is u_x = u * cos(theta), where theta is the angle at which the ball is passed.
- The horizontal component of the players' velocity (v) is v_x = v.

2. Find the vertical components of both velocities:
Since the players are moving straight down the field in the y-direction, both their velocity (v) and the ball's velocity (u) will have no y-component.

3. Equate the horizontal components of the velocities, v_x = u_x:
v = u * cos(theta)

4. Solve for the angle (theta):
theta = arccos(v/u)

Now, let's substitute the given values:

v = 7 m/s
u = 10 m/s

theta = arccos(7/10)

Using a calculator, we can find that the angle is approximately 44.42 degrees. So, the player should pass the ball at an angle of approximately 44.42 degrees with respect to their trajectory.