(Modeling)Solve each problem. See Example 3.Height of a ProjectileA projectile is launched from ground level with an initial velocity of v_0 feet per second. Neglecting air resistance, its height in feet t seconds after launch is given by s=-16t^2+v_0t. 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 v_0. Round answers to the nearest hun-dredth if necessary. v_0=96

Verified step by step guidance

1. Identify the equation for the height of the projectile: \( s = -16t^2 + v_0t \).
2. Substitute the given initial velocity \( v_0 = 96 \) into the equation: \( s = -16t^2 + 96t \).
3. For part (a), set \( s = 80 \) and solve the equation \( -16t^2 + 96t = 80 \) for \( t \).
4. Rearrange the equation to form a standard quadratic equation: \( -16t^2 + 96t - 80 = 0 \).
5. Use the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the values of \( t \), where \( a = -16 \), \( b = 96 \), and \( c = -80 \).

Key Concepts

Quadratic Functions

The height of a projectile can be modeled by a quadratic function, which is a polynomial of degree two. In this case, the equation s = -16t^2 + v_0t represents a parabola that opens downward, indicating that the projectile will rise to a maximum height and then fall back to the ground. Understanding the properties of quadratic functions, such as their vertex and roots, is essential for solving problems related to projectile motion.

Roots of a Quadratic Equation

Finding the time(s) when the projectile reaches a specific height or returns to the ground involves solving for the roots of the quadratic equation. The roots can be found using the quadratic formula, factoring, or completing the square. In this context, the roots represent the times at which the height of the projectile equals 80 feet or zero feet (ground level), which are critical for understanding the projectile's motion.

Projectile Motion

Projectile motion describes the motion of an object that is launched into the air and is influenced only by gravity (neglecting air resistance). The initial velocity, v_0, determines how high and how far the projectile will travel. The equation s = -16t^2 + v_0t captures the effects of gravity on the projectile's height over time, making it essential to analyze the motion accurately.