Question

a student kicks a football vertically upwards, causing the football to reach a maximum height of 43.0m before falling back to the ground. determine all unknowns and answer the following questions. neglect drag and the initial height and horizontal motion of the football. with what speed was the football kicked? m/s what was the footballs total flight time? s

Explanation:

Step1: Use kinematic - equation for vertical motion at max - height

At the maximum height, the final velocity $v = 0$. The kinematic equation $v^{2}=v_{0}^{2}-2gh$ (where $v$ is final velocity, $v_{0}$ is initial velocity, $g = 9.8\ m/s^{2}$ is acceleration due to gravity and $h$ is height) can be used. Rearranging for $v_{0}$ gives $v_{0}=\sqrt{2gh}$.

Step2: Calculate initial velocity

Substitute $g = 9.8\ m/s^{2}$ and $h = 43.0\ m$ into the formula: $v_{0}=\sqrt{2\times9.8\times43.0}=\sqrt{842.8}\approx 29.0\ m/s$.

Step3: Use kinematic - equation for time of flight

The kinematic equation $v = v_{0}-gt$ is used to find the time to reach the maximum - height. At the maximum height $v = 0$. Rearranging for $t$ gives $t=\frac{v_{0}}{g}$. The total flight time $T = 2t$.

Step4: Calculate total flight time

First, find $t=\frac{v_{0}}{g}=\frac{29.0}{9.8}\ s$. Then $T = 2\times\frac{29.0}{9.8}=\frac{58.0}{9.8}\approx 5.92\ s$.

Answer:

Initial speed: 29.0 m/s
Total flight time: 5.92 s