Calculus Question

Question

A stone thrown upward from the top of a 121 ft cliff at 165 ft/sec eventually falls to the beach below. (For this problem take the acceleration due to gravity to be  ft/sec/sec; take upwards to be a positive direction.)

How long does the stone take to reach its highest point?

What is its maximum height?

How long before the stone hits the beach?

What is the velocity of the stone upon impact?

Answer 1

s(t) = gt²/2 + v₀t + s₀ = -16t²+165t+121

a)

To get the maximum height, take the derivative, set that to zero and solve for t

s'(t) = -32t + 165

-32t+165 = 0

t = 165/32 ≈ 5.16 seconds

b)

s(165/32) = -16(165/32)² + 165(165/32) + 121 = 546.39 feet

c) Use the quadratic equation to solve for t in

-16t²+165t+121 = 0

t = -0.7, 11 Take the positive answer

t = 11 seconds

d) v(t) = s'(t)

v(11) = -32(11) + 165 = -187

Velocity = 187 ft/sec in the down direction