Initial velocity Suppose a baseball is thrown vertically upward from the ground with an initial velocity of v0ft/s Its height above the ground after t seconds is given by s(t) = -16t²+v0t. Determine the initial velocity of the ball if it reaches a high point of 128 ft.

Verified step by step guidance

1. Identify the formula for the height of the baseball: \( s(t) = -16t^2 + v_0t \).
2. Recognize that the high point of the ball is reached when its velocity is zero. The velocity function is the derivative of the height function: \( v(t) = \frac{d}{dt}(-16t^2 + v_0t) = -32t + v_0 \).
3. Set the velocity function equal to zero to find the time \( t \) when the ball reaches its highest point: \( -32t + v_0 = 0 \). Solve for \( t \) to get \( t = \frac{v_0}{32} \).
4. Substitute \( t = \frac{v_0}{32} \) into the height function \( s(t) = -16t^2 + v_0t \) and set it equal to 128 ft, the maximum height: \( -16\left(\frac{v_0}{32}\right)^2 + v_0\left(\frac{v_0}{32}\right) = 128 \).
5. Simplify and solve the equation for \( v_0 \) to determine the initial velocity of the ball.

Key Concepts

Quadratic Functions

The height of the baseball is modeled by a quadratic function, s(t) = -16t² + v0t, where the term -16t² represents the effect of gravity. Quadratic functions have a parabolic shape, and their maximum or minimum points can be found using the vertex formula. In this case, the vertex represents the highest point the baseball reaches.

Vertex of a Parabola

The vertex of a parabola given by the equation s(t) = at² + bt + c can be found using the formula t = -b/(2a). This point gives the maximum height when the parabola opens downwards, as in this scenario. Understanding how to find the vertex is crucial for determining the time at which the baseball reaches its highest point.

Initial Velocity

Initial velocity, denoted as v0 in the equation, is the speed at which the baseball is thrown upward. It directly influences how high the baseball will rise before gravity pulls it back down. To find the initial velocity when the baseball reaches a height of 128 ft, we can substitute the height into the height equation and solve for v0.