Answer to Question #133686 in Quantum Mechanics for [e-mail]

Question

Calculate the maximum distance from a goal post that a player must stand if he must shoot the ball to score a goal,assuming the player can shoot the ball with a maximum velocity of 10 metre per second

Answer 1

We may assume an arbitrary angle between the horizontal and the initial velocity vector.

Let the x-axis be directed from the player to the goal post. The x-coordinate after the time t will be

$x(t) = x_0 + v_0\cos\alpha\cdot t = v_0\cos\alpha\cdot t.$

The total time of the flight of a ball will be twice the time of the movement to the maximum height (the point in which the vertical component of the velocity becomes 0, so $v_0\sin\alpha - gt = 0$ ).

$t = 2 \cdot \dfrac{v_0\sin\alpha - 0}{g} = \dfrac{2v_0\sin\alpha}{g}.$

Therefore,

$x(t) = v_0\cos\alpha \cdot \dfrac{2v_0\sin\alpha}{g} = \dfrac{v_0^2\cdot\sin2\alpha}{g}$ .

The maximum value of $\sin2\alpha$ is 1, so the maximum distance will be $\dfrac{v_0^2}{g} \approx 10\,\mathrm{m}.$