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

The position vector of the ball after t seconds is given by the formula:

$ext{Position} = (2i + j) + t(5i + 8j)$

Thus, the position vector becomes:

$(2 + 5t)i + (1 + 8t)j$

The point B has a position vector of (10i + 7j). For the ball to be due north of B, the x-component must match:

$2 + 5t = 10 \\Rightarrow 5t = 8 \\Rightarrow t = 1.6 \, ext{s}$

At time t = 1.6 seconds, the position vector of the ball can be calculated:

$\text{Position} = (2 + 5(1.6))i + (1 + 8(1.6))j = (10i + 13.8j)$

The player starts running north from B, and in order to intercept the ball, we calculate the distance travelled. The player moves due north at speed v:

$\text{Distance travelled by the second player} = 13.8 - 7 = 6.8 \, ext{m}$

Thus, the time taken by the second player is:

$t = \\frac{6.8}{v}$

Since they must meet at the same time:

$1.6 = \frac{6.8}{v} \\Rightarrow v = \frac{6.8}{1.6} = 4.25 \, ext{m s}^{-1}$

One physical factor to consider, other than air resistance, is the effect of wind on the ball's trajectory. Wind can alter the path and speed of the ball, making the model more realistic.