Throwing a stone Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 32 ft/s from a height of 48 ft above the ground. The height (in feet) of the stone above the ground t seconds after it is thrown is $s (t) = - 16 t^{2} + 32 t + 48$ .
d. When does the stone strike the ground?

검증된 단계별 안내

Identify the height function: $ s(t) = -16t^2 + 32t + 48 $.

Set the height function equal to zero to find when the stone hits the ground: $ -16t^2 + 32t + 48 = 0 $.

Use the quadratic formula $ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $ to solve for $ t $, where $ a = -16 $, $ b = 32 $, and $ c = 48 $.

Calculate the discriminant $ b^2 - 4ac $ to determine the nature of the roots.

Solve for $ t $ using the quadratic formula to find the time when the stone strikes the ground.

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주요 개념

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Quadratic Functions

The height of the stone is modeled by a quadratic function, which is a polynomial of degree two. In this case, the function s(t) = -16t² + 32t + 48 represents a parabola that opens downward due to the negative coefficient of t². Understanding the properties of quadratic functions, such as their vertex, axis of symmetry, and roots, is essential for analyzing the motion of the stone.

Finding Roots

To determine when the stone strikes the ground, we need to find the roots of the quadratic equation, which are the values of t for which s(t) = 0. This involves solving the equation -16t² + 32t + 48 = 0, typically using methods such as factoring, completing the square, or applying the quadratic formula. The roots represent the times at which the height of the stone is zero, indicating it has hit the ground.

Projectile Motion

The motion of the stone can be analyzed through the principles of projectile motion, which describes the trajectory of an object under the influence of gravity. In this scenario, the initial velocity and height affect the stone's path. Understanding the effects of gravitational acceleration (approximately -32 ft/s² on Earth) and how it influences the stone's upward and downward motion is crucial for predicting when it will reach the ground.