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).
b. Determine the instantaneous velocity of the projectile at t=1 and t = 2 seconds.
s(t)= −16t²+100t

Verified step by step guidance

1. Step 1: Understand that the instantaneous velocity of a projectile at a given time is the derivative of its position function s(t) with respect to time t. This derivative is denoted as v(t) = s'(t).
2. Step 2: Given the position function s(t) = -16t^2 + 100t, find the derivative s'(t) using the power rule. The power rule states that the derivative of t^n is n*t^(n-1).
3. Step 3: Apply the power rule to each term in s(t). The derivative of -16t^2 is -32t, and the derivative of 100t is 100. Therefore, s'(t) = -32t + 100.
4. Step 4: To find the instantaneous velocity at t = 1 second, substitute t = 1 into the derivative s'(t). Calculate v(1) = -32(1) + 100.
5. Step 5: To find the instantaneous velocity at t = 2 seconds, substitute t = 2 into the derivative s'(t). Calculate v(2) = -32(2) + 100.

Key Concepts

Instantaneous Velocity

Instantaneous velocity is the rate of change of an object's position with respect to time at a specific moment. It is mathematically represented as the derivative of the position function, s(t), with respect to time, t. For a function s(t), the instantaneous velocity at time t is found by calculating s'(t), which gives the slope of the tangent line to the position curve at that point.

Derivative

The derivative is a fundamental concept in calculus that measures how a function changes as its input changes. It provides a way to calculate the slope of the tangent line to the curve of the function at any given point. In the context of the position function s(t), the derivative s'(t) represents the instantaneous velocity of the projectile, allowing us to determine how fast the projectile is moving at any specific time.

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form s(t) = at² + bt + c, where a, b, and c are constants. In this case, the position function s(t) = -16t² + 100t describes the motion of a projectile under the influence of gravity. Understanding the properties of quadratic functions, such as their parabolas and vertex, is essential for analyzing the motion of the projectile and determining its velocity at specific times.