31–32. Velocity functions A projectile is fired vertically upward into the air, and its position (in feet) above the ground after t seconds is given by the function s(t).
a. For the following functions s(t), find the instantaneous velocity function v(t). (Recall that the velocity function v is the derivative of the position function s.)
s(t)= −16t²+100t

Verified step by step guidance

1. Step 1: Identify the position function s(t) given in the problem, which is s(t) = -16t^2 + 100t.
2. Step 2: Recall that the instantaneous velocity function v(t) is the derivative of the position function s(t) with respect to time t.
3. Step 3: Differentiate the position function s(t) = -16t^2 + 100t with respect to t. Use the power rule for differentiation, which states that the derivative of t^n is n*t^(n-1).
4. Step 4: Apply the power rule to each term in s(t). The derivative of -16t^2 is -32t, and the derivative of 100t is 100.
5. Step 5: Combine the derivatives to find the velocity function v(t). Therefore, v(t) = -32t + 100.

Key Concepts

Derivative

The derivative of a function measures how the function's output changes as its input changes. In the context of motion, the derivative of the position function s(t) gives the instantaneous velocity v(t). This concept is fundamental in calculus as it provides a way to analyze rates of change.

Instantaneous Velocity

Instantaneous velocity refers to the velocity of an object at a specific moment in time. It is calculated as the derivative of the position function with respect to time. For the given position function s(t), finding v(t) involves applying differentiation to determine how fast the projectile is moving at any time t.

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form s(t) = at² + bt + c. In this case, the position function s(t) = -16t² + 100t is a quadratic function, where the coefficients determine the shape of the parabola. Understanding the properties of quadratic functions is essential for analyzing projectile motion.