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Chapter 3 | Derivatives
Example 3.34 Comparing Instantaneous Velocity and Average Velocity
A ball is dropped from a height of 64 feet. Its height above ground (in feet) t seconds later is given by s ( t ) =−16 t 2 +64.
a. What is the instantaneous velocity of the ball when it hits the ground? b. What is the average velocity during its fall?
Solution The first thing to do is determine how long it takes the ball to reach the ground. To do this, set s ( t ) =0. Solving −16 t 2 +64=0, we get t =2, so it take 2 seconds for the ball to reach the ground. a. The instantaneous velocity of the ball as it strikes the ground is v (2). Since v ( t ) = s ′( t ) =−32 t , we obtain v ( t ) = −64 ft/s. b. The average velocity of the ball during its fall is v ave = s (2)− s (0) 2−0 = 0−64 2 = −32 ft/s.
Example 3.35 Interpreting the Relationship between v ( t ) and a ( t )
A particle moves along a coordinate axis in the positive direction to the right. Its position at time t is given by s ( t ) = t 3 −4 t +2. Find v (1) and a (1) and use these values to answer the following questions. a. Is the particle moving from left to right or from right to left at time t =1? b. Is the particle speeding up or slowing down at time t =1?
Solution Begin by finding v ( t ) and a ( t ).
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