Question

A soccer player uses her head to hit a ball up in the air from a height of 1 meters with an initial vertical velocity of 5 meters per second. The height h in meters of the ball is given by h = −4.9t2 + 5t + 1, where t is the time elapsed in seconds. How long will it take the ball to hit the ground if no other players touch it? Enter the time to two decimal places.

Answer 1

It will take 1.19 seconds for the ball to hit the ground if no other players touch it

Explanation:

The height of the ball after t seconds is given by the following equation:

`h(t) = -4.9t^(2) + 5t + 1`

How long will it take the ball to hit the ground if no other players touch it?

This is t when h(t) = 0. So

`-4.9t^(2) + 5t + 1 = 0`

Solving a quadratic equation:

Given a second order polynomial expressed by the following equation:

`ax^(2) + bx + c, a\\eq0`.

This polynomial has roots
`x_(1), x_(2)` such that
`ax^(2) + bx + c = a(x - x_(1))*(x - x_(2))`, given by the following formulas:

`x_(1) = (-b + √(\bigtriangleup))/(2*a)`

`x_(2) = (-b - √(\bigtriangleup))/(2*a)`

`\bigtriangleup = b^(2) - 4ac`

In this question:

`-4.9t^(2) + 5t + 1 = 0`

So
`a = -4.9, b = 5, c = 1`

So

`\bigtriangleup = 5^(2) - 4*4.9*1 = 44.6`

`t_(1) = (-5 + √(44.6))/(2*(-4.9)) = -0.17`

Since we want a time measure, the answer cannot be negative.

It will take 1.19 seconds for the ball to hit the ground if no other players touch it