Chapte 2.3-108
The position function f ( t ) = −16t 2 + 1000 gives the height of an object that has fallen for t seconds from a height of 1000 feet. The velocity at time t = a seconds is given by lim
t →a
s ( a ) − s (t )
. a −t
If a construction worker drops a wrench from a height of 1000 feet, when will the wrench hit the ground? At what velocity will the wrench impact the ground? Answers and Solutions: The wrench is on the ground when −16t 2 + 1000 = 0 when t 2 =
1000
10 10 5
or t =
=
10 16
4
2
seconds. To find the velocity of the wrench as it hits the ground, we want to know the velocity as t→
5
10 . 2
Replace a with 5
s ( a ) − s (t )
. 10 in lim
t →a
2
a −t
5
s ⎛⎜ 10 ⎞⎟ − s ( t )
⎠
lim ⎝ 2
5
5
t → 10
10 − t
2
2
2
⎡
⎛5
⎞ + 1000 ⎤ − [
2
16
−
10
⎜
⎟
⎢
⎥ −16t + 1000]
⎝2
⎠
⎦
lim ⎣
5
5
t → 10
10 − t
2
2
⎡
⎤
⎛ 1000 ⎞
2
⎢ −16 ⎜⎝ 16 ⎟⎠ + 1000 ⎥ − [ −16t + 1000]
⎣
⎦
lim
5
5
t → 10
10 − t
2
2
lim
t→
[ 0] + 162 − 1000
5
10 − t
2
5
10
2
5
10 + t
162 − 1000 2
lim
⋅
5
5
t → 10 5
10 + t
10 − t
2
2
2
(162 − 1000 ) ⎛⎜
lim
t→
5
10
2
5
⎞
10 + t ⎟
⎝2
⎠
250 2
−t
4
5
4 ( 4t 2 − 250 ) ⎛⎜ 10 + t ⎞⎟
⎝2
⎠
lim
5
250
t → 10
− t2
2
4
5
4 ( 4t 2 − 250 ) ⎛⎜ 10 + t ⎞⎟
⎝2
⎠⋅4
lim
5
2
50
4
t → 10
− t2
2
4
5
16 ( 4t 2 − 250 ) ⎛⎜ 10 + t ⎞⎟
⎝2
⎠
lim
5
2
( 250 − 4t )
t → 10
2
5
5
5
⎞
lim − 16 ⎛⎜ 10 + t ⎞⎟ = −16 ⎛⎜ 10 +
10 ⎟ = −80 10
5
2
⎝2
⎠
⎝2
⎠
t → 10
2
The velocity when the wrench hits the ground is approximately 253 feet per second. The reason the velocity is negative is because the wrench is falling.