Question

a goalie punts the soccer ball down the field. the soccer ball has an initial velocity of 20 m/s horizontally and an initial velocity of 15 m/s vertically. a). find the total time. b). what is the maximum height that the ball reaches?

Explanation:

Step1: Find time to reach maximum height

Use the vertical - motion formula $v = v_0+at$. At the maximum height, the vertical final velocity $v_y = 0$. The acceleration due to gravity $a=-g=- 9.8\ m/s^{2}$, and the initial vertical velocity $v_{0y}=15\ m/s$. Rearranging the formula for time $t=\frac{v - v_0}{a}$, we get $t_{up}=\frac{0 - 15}{-9.8}\ s$.
$t_{up}=\frac{- 15}{-9.8}\ s\approx1.53\ s$

Step2: Calculate total time

The time of flight $t_{total}$ is twice the time to reach the maximum height. So $t_{total}=2t_{up}$.
$t_{total}=2\times1.53\ s = 3.06\ s$

Step3: Find maximum height

Use the formula $v^{2}-v_{0}^{2}=2ad$. In the vertical - direction, $v_y = 0$ at the maximum height, $v_{0y}=15\ m/s$, and $a=-9.8\ m/s^{2}$. Rearranging for displacement $d = y_{max}$, we have $y_{max}=\frac{v_{y}^{2}-v_{0y}^{2}}{2a}$.
$y_{max}=\frac{0 - 15^{2}}{2\times(-9.8)}=\frac{-225}{-19.6}\ m\approx11.48\ m$

Answer:

a. $t_{total}=3.06\ s$
b. $y_{max}\approx11.48\ m$