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) = -16t² + 32t + 48.
With what velocity does the stone strike the ground?

검증된 단계별 안내

Step 1: Understand the problem. We need to find the velocity of the stone when it strikes the ground. The height function is given by s(t) = -16t^2 + 32t + 48, where s(t) is the height in feet and t is the time in seconds.

Step 2: Determine when the stone hits the ground. This occurs when s(t) = 0. Solve the equation -16t^2 + 32t + 48 = 0 to find the time t when the stone reaches the ground.

Step 3: Use the quadratic formula to solve for t. The quadratic formula is t = (-b ± √(b^2 - 4ac)) / (2a), where a = -16, b = 32, and c = 48.

Step 4: Once you have the value of t when the stone hits the ground, find the velocity at that time. The velocity function v(t) is the derivative of the height function s(t).

Step 5: Differentiate s(t) to find v(t). The derivative of s(t) = -16t^2 + 32t + 48 is v(t) = ds/dt = -32t + 32. Substitute the value of t from Step 3 into v(t) to find the velocity 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, s(t) = -16t² + 32t + 48. Quadratic functions are polynomial functions of degree two, characterized by their parabolic shape. Understanding how to analyze and manipulate these functions is crucial for determining the stone's height at any given time and finding when it reaches the ground.

Velocity and Acceleration

Velocity is the rate of change of position with respect to time, and it can be derived from the height function by taking its first derivative, s'(t). In this context, the stone's velocity will change due to gravitational acceleration, which is represented by the coefficient of the t² term in the height equation. Recognizing how to compute and interpret velocity is essential for determining how fast the stone is moving when it strikes the ground.

Roots of a Function

Finding the roots of the height function s(t) is necessary to determine when the stone hits the ground, which occurs when s(t) = 0. The roots can be found using the quadratic formula or factoring, and they represent the time(s) at which the height of the stone is zero. This concept is fundamental in solving problems involving motion and understanding the behavior of quadratic equations.