Calculus velocity

Question

In a scene from the Martin Matt Damon shoots a flare straight up into the sky. The height( in feet) of the flare after t seconds is given by

s(t)= -3t squared + 100t + 3

What is the velocity of the flare when it is moving toward the ground and its position is equal to 131 feet?

Answer 1

We first solve for t, then find the velocity at this time

131 = -3t^2 + 100t + 3

-3t^2 + 100t - 128 = 0

-(3t - 4)(t - 32) = 0

t = 4/3, 32

The flare will be going down at 32

s'(t) = -6t + 100

s'(32) = -6(32) +100 = -92

The velocity is 92 ft/sec down.

Answer 2

Start by finding the time when the flare is moving toward the ground and at a height of 131 feet (there will be two solutions, we will take the second one).

131= -3t²+100t+3

-3t²+100t-128=0

Use quadratic formula to find the times, choose the larger time (after the flare has begun to descend)

t1=4/3 seconds, t2=32 seconds

t=t2=32 seconds

Now take the derivative of s(t) with respect to t and plug in 32 seconds for t.

ds/dt= -6t+100

ds/dt(32)= -6(32)+100 = -92 feet/second