At time t = 0, a football player kicks a ball from the point A with position vector (2i + j) m on a horizontal football field - Edexcel - A-Level Maths Mechanics - Question 8 - 2005 - Paper 1

To find the position vector of the ball after t seconds, we use the initial position vector and the velocity vector:
$\text{Position vector} = (2i + j) + t(5i + 8j)$
Hence, the position vector after t seconds is:
$\text{Position vector} = [(2 + 5t)i + (1 + 8t)j]$

To find when the ball is due north of B, we set the x-coordinate of the ball's position equal to the x-coordinate of point B, since due north means the same x-coordinate.
From B, we have:
$10 = 2 + 5t$
Solving for t gives:
$5t = 8 \Rightarrow t = 1.6 \, \text{s}$

At t = 1.6 s, the position vector of the ball is:
$\text{Position vector} = [2 + 5(1.6)]i + [1 + 8(1.6)]j = [10i + 13.8j]$
Since the player starts from B at (10i + 7j) and moves directly north, the distance traveled by the player in 1.6 s must equal the distance the ball has moved vertically.
The distance traveled by the second player:
$\text{Distance} = 13.8 - 7 = 6.8 \Rightarrow r \times 1.6 = 6.8 \Rightarrow r = \frac{6.8}{1.6} = 4.25 \, \text{m s}^{-1}$

One physical factor that could be taken into account for a more realistic model of the ball's motion is 'wind,' as it can affect the trajectory and speed of the ball significantly.