Hello, mj.alawami!

A boy kicked a football with an initial vertical velocity component of 15.0 m/s

and a horizontal velocity component of 22.0 m/s.

a) What is the velocity of the football (magnitude and direction)?

The velocity can be represented by this triangle: Code:

*
* |
* | 15
* θ |
* - - - - - - - *
22

The magnitude of the velocity is the length of the hyptenuse.

. . $\displaystyle |v| \:=\:\sqrt{22^2 + 15^2} \:=\:\sqrt{709} \:\approx\:26.6$ m/s.

The direction of the velocity is given by:

. . $\displaystyle \tan\theta \:=\:\frac{15}{22} \quad\Rightarrow\quad \theta \:=\:\arctan\left(\tfrac{15}{22}\right) \:\approx\:34.3^o$

b) How much time is needed to reach the maximum height?

The height $\displaystyle y$ of the football is given by: .$\displaystyle y \:=\:15t - 4.9t^2$

The graph is a down-opening parabola which reaches its maximum at its vertex.

The vertex is at: .$\displaystyle t \:=\:\frac{\text{-}b}{2a} \:=\:\frac{\text{-}15}{2(\text{-}4.9)} \:=\:1.530612245$

The ball reaches maximum height in abut 1.5 seconds.