Height of a Projectile A projectile is launched from ground...

Height of a Projectile A 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 launchis given by$$s=-16 t^{2}+v_{0} t$$ Find the time(s) that the projectile will (a) reach a height of 80 $\mathrm{ft}$. and (b) return to the ground for the given value of $v_{0}$. Round answers to the nearest hundredth if necessary. $v_{0}=96$

Height of a Projectile A 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 launchis given by$$s=-16 t^{2}+v_{0} t$$
Find the time(s) that the projectile will (a) reach a height of 80 $\mathrm{ft}$. and (b) return to the ground for the given value of $v_{0}$. Round answers to the nearest hundredth if necessary.

$v_{0}=96$

Key Concepts

Projectile Motion

Projectile motion refers to the motion of an object thrown into the air and moving under the influence of gravity alone, without propulsion. It is characterized by a parabolic trajectory and involves the decomposition of the initial velocity into horizontal and vertical components. This concept is fundamental in physics for analyzing the behavior of objects in free-fall under uniform acceleration due to gravity.

Kinematic Equations Under Constant Acceleration

Kinematic equations relate displacement, initial velocity, time, and acceleration for objects in motion under constant acceleration. In projectile motion, these equations are used to analyze the vertical component of the motion, typically incorporating the gravitational acceleration, which is constant and creates a quadratic relation between height and time.

Quadratic Equations

Quadratic equations are second-degree polynomial equations that often arise in the study of projectile motion when expressing the height of a projectile as a function of time. The solutions to these equations, typically found using the quadratic formula, represent the times at which the projectile reaches specific heights or returns to the ground.

Discriminant and Nature of Roots

The discriminant of a quadratic equation, derived from the quadratic formula, indicates the number and type of solutions (roots) to the equation. In the context of projectile motion, a positive discriminant suggests two real solutions, corresponding to two distinct moments in time when the projectile reaches a specific height, while a discriminant of zero indicates a single solution, and a negative discriminant would indicate no real solution.

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