Question

12 at a football game a t - shirt is launched straight up into the air and is in the air for 9.0 s. what will be its maximum height?
a 99.23 m
b 396.9 m
c 9 m
d 88.2 m

Explanation:

Step1: Find the time to reach maximum height

The total time of flight \(t = 9.0\ s\). For vertical - motion under gravity (symmetric motion), the time to reach the maximum height \(t_{up}\) is half of the total time of flight. So \(t_{up}=\frac{t}{2}\).
Substituting \(t = 9.0\ s\), we get \(t_{up}=\frac{9.0}{2}=4.5\ s\).

Step2: Use the kinematic equation \(h = v_0t-\frac{1}{2}gt^{2}\). At the maximum height, the final velocity \(v = v_0 - gt_{up}=0\), so \(v_0=gt_{up}\).

We can also use the equation \(h=\frac{1}{2}gt_{up}^{2}\) (derived from \(v^{2}-v_0^{2}=- 2gh\) with \(v = 0\) and \(v_0 = gt_{up}\)).
Taking \(g = 9.8\ m/s^{2}\) and \(t_{up}=4.5\ s\), then \(h=\frac{1}{2}\times9.8\times(4.5)^{2}\).
First, calculate \((4.5)^{2}=20.25\). Then \(\frac{1}{2}\times9.8\times20.25 = 4.9\times20.25\).
\(4.9\times20.25=(5 - 0.1)\times20.25=5\times20.25-0.1\times20.25=101.25 - 2.025 = 99.225\approx99.23\ m\).

Answer:

A. \(99.23\ m\)