Height of a Projectile A projectile is launched from ground level with an initial velocity of v₀ feet per second. Neglecting air resistance, its height in feet t seconds after launch is given by s=-16t²+v₀t. In each exercise, find the time(s) that the projectile will (a) reach a height of 80 ft and (b) return to the ground for the given value of v₀. Round answers to the nearest hundredth if necessary. v₀=32

Verified step by step guidance

1. Start with the given height function for the projectile: \(s = -16t^2 + v_0 t\), where \(v_0 = 32\) feet per second. Substitute \(v_0\) into the equation to get \(s = -16t^2 + 32t\).
2. For part (a), set the height \(s\) equal to 80 feet to find the time(s) when the projectile reaches this height: \(-16t^2 + 32t = 80\).
3. Rearrange the equation to standard quadratic form: \(-16t^2 + 32t - 80 = 0\). To simplify, divide the entire equation by -16, resulting in \(t^2 - 2t + 5 = 0\).
4. Use the quadratic formula to solve for \(t\): \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=1\), \(b=-2\), and \(c=5\). Calculate the discriminant \(b^2 - 4ac\) to determine the nature of the roots.
5. For part (b), find the time when the projectile returns to the ground by setting \(s=0\): \(-16t^2 + 32t = 0\). Factor the equation and solve for \(t\) to find the time(s) when the height is zero.

Key Concepts

Quadratic Functions and Their Graphs

A quadratic function is a polynomial of degree two, typically written as f(t) = at^2 + bt + c. Its graph is a parabola that opens upward if a > 0 and downward if a < 0. In projectile motion, the height function is quadratic, representing the path of the projectile over time.

Solving Quadratic Equations

To find specific times when the projectile reaches a certain height, you solve quadratic equations by setting the height function equal to that value. Methods include factoring, completing the square, or using the quadratic formula, which provides the roots (solutions) representing time values.

Interpreting Solutions in Context

The solutions to the quadratic equation correspond to meaningful times: when the projectile reaches a given height and when it returns to the ground (height zero). Negative time values are not physically meaningful here, so only positive solutions are considered, and rounding is applied as needed.