SECTION 14.4 The Tangent Problem; The Derivative 965 Finding the Instantaneous Velocity of an Object In physics it is shown that the height s of a ball thrown straight up with an initial velocity of 80 feet per second ft sec ( ) from a rooftop 96 feet high is ( ) = = − + + s s t t t 16 80 96 2 where t is the elapsed time that the ball is in the air. The ball misses the rooftop on its way down and eventually strikes the ground. See Figure 20(a). (a) When does the ball strike the ground? That is, how long is the ball in the air? (b) At what time t will the ball pass the rooftop on its way down? (c) What is the average velocity of the ball from = t 0 to = t 2? (d) What is the instantaneous velocity of the ball at time t0? (e) What is the instantaneous velocity of the ball at = t 2? (f) When is the instantaneous velocity of the ball equal to zero? (g) What is the instantaneous velocity of the ball as it passes the rooftop on the way down? (h) What is the instantaneous velocity of the ball when it strikes the ground? Solution EXAMPLE 6 (a) The ball strikes the ground when ( ) = = s s t 0. ( )( ) − + + = − − = − + = = = − t t t t t t t t 16 80 96 0 5 6 0 6 1 0 6 or 1 2 2 Discard the solution = − t 1. The ball strikes the ground after 6 sec. (b) The ball passes the rooftop when ( ) = = s s t 96. ( ) − + + = − = − = = = t t t t t t t t 16 80 96 96 5 0 5 0 0 or 5 2 2 Discard the solution = t 0. The ball passes the rooftop on the way down after 5 sec. (c) The average velocity of the ball from = t 0 to = t 2 is ( ) ( ) Δ Δ = − − = − = s t s s 2 0 2 0 192 96 2 48 ft sec (d) The instantaneous velocity of the ball at time t0 is the derivative ( ) ′s t ; 0 that is, ( ) [ ] ( ) ( ) ( )( ) ( ) [ ] ( )( ) ( ) [ ] ( ) ( ) ( ) ′ = − − = − + + − − + + − = − − − + − = − + − − − − = − ⎡⎣ + − − ⎤⎦ − = − + − =− − → → → → → → s t s t s t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t lim lim 16 80 96 16 80 96 lim 16 5 5 lim 16 5 lim 16 5 lim 16 5 162 5 ftsec t t t t t t t t t t t t 0 0 0 2 0 2 0 0 2 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Figure 20(a) 96 ft Roof top (continued)
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