The equation s=-16 t^2+v0 t+s0 gives the height s, in feet...

The equation $$s=-16 t^{2}+v_{0} t+s_{0}$$ gives the height $s$, in feet above ground level, of an object t seconds after the object is thrown directly upward from a height $s_{0}$ feet above the ground with an initial velocity of $v_{0}$ feet per second. A ball is thrown directly upward from ground level with an initial velocity of 64 feet per second. Find the time interval during which the ball has a height of more than 48 feet.

The equation
$$s=-16 t^{2}+v_{0} t+s_{0}$$
gives the height $s$, in feet above ground level, of an object t seconds after the object is thrown directly upward from a height $s_{0}$ feet above the ground with an initial velocity of $v_{0}$ feet per second. A ball is thrown directly upward from ground level with an initial velocity of 64 feet per second. Find the time interval during which the ball has a height of more than 48 feet.

Key Concepts

Projectile Motion

Projectile motion describes the motion of an object thrown into the air subject only to the acceleration due to gravity. It is typically modeled with a quadratic equation that gives the object's height as a function of time. This concept is central in kinematics, where initial velocity, launch angle, and gravitational acceleration determine the trajectory of the projectile.

Quadratic Functions

Quadratic functions are mathematical expressions of the form ax² + bx + c and are characterized by their parabolic graphs. These functions are used to model situations where the rate of change itself changes at a constant rate, such as the height of a projectile over time. The vertex of the parabola represents the maximum or minimum value, which is crucial in analyzing the peak of a projectile's trajectory.

Quadratic Inequalities

Quadratic inequalities involve determining the set of variable values for which a quadratic expression is greater than or less than a given number. In the context of projectile motion, solving a quadratic inequality allows us to find the time intervals during which the height of the projectile exceeds or falls below a certain value. The solution typically involves finding the roots of the corresponding quadratic equation and testing intervals between these roots.

Quadratic Formula

The quadratic formula, given by (-b ± ?(b² - 4ac))/(2a), is a method for finding the roots or zeros of a quadratic equation. This formula is essential in solving quadratic inequalities as it provides the boundary points where the projectile reaches specific heights. Knowledge of these roots enables the determination of intervals where the function satisfies the inequality.

Transcript

00:01 So in this problem, we're told that the equation s equals negative 16t squared plus v .0 t plus s 0 represents the height in feet above the ground of an object t seconds after is thrown directly upwards, where s of 0 is the initial height and v .0 is the initial velocity.

00:18 So we're now told that a ball is thrown directly upward from ground level with an initial velocity of 64 p p for second.

00:27 So that means v .0 would be 64.

00:30 We want to find the time interval during which the ball has a height of more than 48 feet, meaning that s is going to be greater than 48.

00:40 And they told us that the ball is thrown directly upward from ground level.

00:46 Well, that would mean that our initial height, s of zero, would equal to zero.

00:50 Okay, so let's substitute these values into our formula.

00:53 Well, we need s to be greater than 48, meaning that this expression, negative 16t squared, plus v -0, which is 64, plus 0, because that plus 64t, i apologize, plus zero, this needs to be greater than 48.

01:11 Okay.

01:12 Well, we have a quadratic inequality.

01:14 So the first thing we need to do is find the zeros.

01:16 So i'm going to subtract 48 from both sides first.

01:19 So we have negative 16t squared plus 64t minus 48.

01:25 And we're trying to figure out when is this expression greater than zero? well, before we actually find our zeros, i notice that each term is divisible by negative 16.

01:34 So i'm going to subtract or divide both sides of our inequality by negative 16.

01:38 So that will leave us with t squared minus 14 plus three.

01:44 But don't forget, when we divide by a negative, we're going to switch our inequality.

01:48 So this is going to be less than zero.

01:50 So this is the inequality that we're essentially trying to solve.

01:54 Because we're trying to find our zeros first, we're going to set this equal to zero.

01:59 So we'll have t squared minus 4t plus 3 equal to 0, and i'm going to solve by factor.

02:06 Because our leading coefficient is 1, we know both factors will start with t.

02:10 So we just have to find two numbers that multiply the 3 that will add to negative 4...

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