Question

A foot ball punter wants to kick the ball so that it hits the ground 50 m from where it is kicked and so that the ball is in the air 4 seconds. at what angle and with what initial speed should the ball be kicked? assume that the ball leaves the kicker's foot from an elevation of 1 meter

Answer 1

Answer on Question #39938 - Physics - Other

1. A foot ball punter wants to kick the ball so that it hits the ground $50\mathrm{m}$ from where it is kicked and so that the ball is in the air 4 seconds. At what angle and with what initial speed should the ball be kicked? Assume that the ball leaves the kicker's foot from an elevation of 1 meter.

In vertical direction the movement is with the constant acceleration $g$ :

$$y = h + v _ {0} \sin \alpha \cdot t - \frac {g t ^ {2}}{2}.$$

At moment $t = 4s$, the coordinates will be $x = l$, $y = 0$.

Let combine two equations into the system:

$$\left\{ \begin{array}{l} l = v _ {0} \cos \alpha \cdot t \\ 0 = h + v _ {0} \sin \alpha \cdot t - \frac {g t ^ {2}}{2} \end{array} , \right. \left\{ \begin{array}{l} v _ {0} = \frac {l}{t \cos \alpha} \\ 0 = h + \frac {l}{t \cos \alpha} \sin \alpha \cdot t - \frac {g t ^ {2}}{2} \end{array} , \right. \left\{ \begin{array}{l} \boxed {\alpha = \arctan \frac {g t ^ {2}}{2} - h} \\ v _ {0} = \frac {l}{t \cos \alpha} \end{array} . \right.$$

Let check the dimensions.

$$\left[ \alpha \right] = \arctan \frac {m}{m} = r a d, \quad \left[ v _ {0} \right] = \frac {m}{s}.$$

Let evaluate the quantities.

$$\alpha = \arctan \frac {9.8 \cdot 4 ^ {2} - 1}{50} = 1.00 (rad) = 57.1 ^ {(0)}, \quad v _ {0} = \frac {50}{4 \cdot \cos 57 ^ {0}} = 23 \left(\frac {m}{s}\right).$$

Answer: 1.00 rad, $23\frac{m}{s}$.