The height s in feet of a ball above the ground at t seconds is given by s=-16 t^2+40 t+100 (a)

step 1 So given position, we can find the instantaneous velocity because that's the derivative of posit

The height s in feet of a ball above the ground at t seconds...

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Key Concepts

Critical Points and Zero Velocity

Finding when the instantaneous velocity is zero involves setting the derivative equal to zero and solving for the time variable. This critical point often corresponds to key moments in the motion, such as when an object reaches its highest or lowest point in a trajectory. For quadratic position functions, this process typically results in solving a linear equation derived from the differentiation.

Differentiation

Differentiation is the mathematical process of finding the derivative of a function, which represents the instantaneous rate of change of the function with respect to one of its variables. In the context of motion, differentiating a position function with respect to time provides the velocity function, giving the rate at which the position changes over time.

Instantaneous Velocity

Instantaneous velocity is defined as the derivative of the position function at a specific point in time. It represents the speed and direction of an object's motion at that exact moment, rather than an average over an interval.

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