Use the position function s(t) = –16t2 + 850, which gives the height (in feet) of an object that has fallen fo?

Question

object that has fallen for t seconds from a height of 700 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 700 feet, how fast will the wrench be falling after 3 seconds?
Use position function s(t)=-16t^2+800 which gives hieght in feet of object that has fallen for t seconds from a height of 800 feet. The velocity at time t=a seconds is given by lim t-a s(a)-s(t)/a-t
1. If construction worker drops wrench from a height of 800 feet, when will wrench hit the ground?
2. At what velocity will wrench impact the ground?

Answer 1

The position function for a free-falling object is given by



s(t) = - 16t² + v0t + h0



where h = height in ft., t = time in secs., v0 = initial velocity in ft./sec. and h0 = initial height in ft..



v0 = 0 ft./sec.

h0 = 800 ft.



Position function for this problem:



s(t) = - 16t² + 800



1.) Time to Impact: s(t) = 0:



- 16t² + 800 = 0

- 16t² = - 800

t = - 800 / - 16

t = 50



Time to Impact = 50 secs.

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2.) The velocity is given by the derivative, s'(t), of the position function at time t:



s'(50) = - 32(50)

s'(50) = - 1600



Velocity at Impact = - 1600 ft./sec. (Negative because the wrench is moving downward)

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Answer 2

1. "Hit the ground" means s = 0. Solve for the value of t when s = 0. It's a very simple quadratic equation, -16t^2 + 800 = 0.



2. I guess you're supposed to use the limit definition of derivative to find velocity.



s(a) = -16a^2 + 800

s(t) = -16t^2 + 800



[s(a) - s(t)]/(a - t) = (-16a^2 + 16t^2)/(a - t) = 16(t^2 - a^2)/(a - t)



Since the numerator is a difference of squares, this is very easy to factor and simplify. Do that and take the limit as a->t.



That's v(t). Plug in the value of t from part a.

Answer 3

Take the 2d time by-product. this provides the acceleration. right here that's -10 m/s/s Use the relation s -s0 - a million/2 a t^2 s=0 so=180m a =10 m/s^2 This supply t=6 the wonderful speed of fall v =a t = 60ms