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=96

Verified step by step guidance

1. Start with the given height function for the projectile: \(s = -16t^2 + v_0 t\), where \(v_0 = 96\) feet per second. Substitute \(v_0\) into the equation to get \(s = -16t^2 + 96t\).
2. For part (a), set the height \(s\) equal to 80 feet to find the time(s) when the projectile reaches this height: \(-16t^2 + 96t = 80\).
3. Rearrange the equation to standard quadratic form: \(-16t^2 + 96t - 80 = 0\). You can simplify this equation by dividing all terms by -16 to make calculations easier.
4. Solve the quadratic equation for \(t\) using the quadratic formula: \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a\), \(b\), and \(c\) are the coefficients from the quadratic equation.
5. For part (b), find the time when the projectile returns to the ground by setting \(s = 0\) and solving \(-16t^2 + 96t = 0\). 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 s(t) = at^2 + bt + c. Its graph is a parabola, which opens upward if a > 0 and downward if a < 0. In projectile motion, the height function is quadratic, modeling the path of the projectile over time.

Solving Quadratic Equations

To find specific times when the projectile reaches a certain height, you solve the quadratic equation s(t) = given height. This involves rearranging the equation to standard form and using methods like factoring, completing the square, or the quadratic formula to find real solutions for t.

Interpreting Solutions in Context

The solutions to the quadratic equation represent times when the projectile is at a certain height. Positive real solutions correspond to meaningful times after launch, while negative or complex solutions are not physically relevant. Understanding this helps identify when the projectile reaches a height or returns to the ground.