Question

A football player kicks a football from a height of 4 feet with an initial vertical velocity of 64 feet per second. Use the vertical motion model, h = –16t^2 + vt +s, where v is the initial velocity in feet per second and s is the height in feet, to calculate the amount of time the football is in the air before it hits the ground. Round your answer to the nearest tenth if necessary.

Answer 1

`4.1s`

Step-by-step explanation

We want to calculate the time it will take for the football to hit the ground.

To do this, we have to solve the equation given for h = 0:

`-16t^2+vt+s=0`

Substitute the given values of v and s into the equation and solve for t:

`-16t^2+64t+4=0`

Solve this using the quadratic formula:

`t=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}`

where:

`a=-16;b=64;c=4`

Therefore:

`\begin{gathered} t=\frac{-64\pm\sqrt[]{64^2-(4\cdot-16\cdot4)}}{2(-16)}=\frac{-64\pm\sqrt[]{4096+256_{}}}{-32} \\ t=\frac{-64\pm\sqrt[]{4352}}{-32}=(-64\pm65.97)/(-32) \\ t=(-64+65.97)/(-32);t=(-64-65.97)/(-32) \\ t=(1.97)/(-32);t=(-129.97)/(-32) \\ \Rightarrow t\approxeq-0.1s;t\approxeq4.1s \end{gathered}`

Since time cannot be negative, it implies that the football will spend 4.1 seconds in the air before it hits the ground.