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Velocity and Acceleration

Velocity and Acceleration. Chapter 2. Rates of Change.

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Velocity and Acceleration

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  1. Velocity and Acceleration Chapter 2

  2. Rates of Change • The derivative can be used to determine the rate of change in one variable with respect to another. Applications involving rates of change occur in the following examples: population growth rates, water flow rates, velocity and acceleration. • A common use for rate of change is to describe the motion of an object moving in a straight line.

  3. The function s that gives the position (relative to the origin) of an object as a function of time t is called a position function. If, over a period of time ∆t, the object changes its position by the amount then by the familiar formula the average velocity is

  4. If a billiard ball is dropped from a height of 100 feet, its height s at time t is given by the position function s(t) = -16t2 + 100 where s is measured in feet and t is measured in seconds. Find the average velocity over the interval [1, 2].

  5. The average velocity between t1and t2 is the slope of the secant line, and the instantaneous velocity at t1is the slope of the tangent line. • If s = s(t) is the position function for an object along a straight line, the velocity of the object at time t is

  6. The position of a free-falling object (neglecting air resistance) under the influence of gravity can be represented by the equation s(t)=½gt2 + v0t + s0 where s0 is the initial height of the object, v0 is the initial velocity of the object, and g is the acceleration due to gravity.

  7. At time t=0, a diver jumps from a platform diving board that is 32 feet above the water. The position of the diver is given by s(t)= -16t2 + 16t +32 where s is measured in feet and t is measured in seconds. • When does the diver hit the water? • What is the diver’s velocity at impact?

  8. A silver dollar is dropped from the top of a building that is 1362 feet tall. • Determine the position, velocity and acceleration functions for the coin. • Determine the average velocity on the interval [1,2]. • Find the instantaneous velocities at t=1 and t=2. • Find the time required for the coin to reach the ground. • Find the velocity of the coin at impact. • Find the acceleration at t=1.2

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