A toy rocket is launched

Question

If a toy rocket is launched vertically upward from ground level with an initial velocity of 128 feet per second, then it's height h after t seconds is given by the equation h(t) =-16t²+128t(if air resistance is neglected)

a) how long will it take for the rocket to return to the ground?

b)after how many seconds will the rocket be 112 feet above the ground?

c)how long will it take the rocket to hit max height?

d)what is max height?

Answer 1

The first question asks how many seconds will go by before the rocket hits the ground. That means you're interested in when the rocket's height h(t) = 0. Factor the equation to find its zeroes:

 

h(t) = -16t² + 128t

h(t) = t(-16t + 128)

0 = t(-16t + 128)

0 = t    OR    0 = -16t + 128

0 = t    OR    -128 = -16t

0 = t    OR    -128/-16 = t

0 = t    OR    8 = t

 

That means that at height = h(t) = 0, t = 0 or 8. At t = 0, zero seconds have gone by and the rocket hasn't yet left the ground, so t = 8 is when the rocket will return to the ground. That means the rocket will take 8 seconds to return to the ground.

 

 

To find out how many seconds it will take the rocket to reach 112 feet, plug in h(t) = 112, rearrange into quadratic form, and solve:

 

112 = -16t² + 128t

0 = -16t² + 128t - 112

0 = -16(t² - 8t + 7)

0 = t² - 8t + 7

0 = (t - 7)(t - 1)

0 = t - 7    OR    0 = t - 1

7 = t    OR    1 = t

 

That means the rocket will be at height = h(t) = 112 at t = 7 and t = 1. That means the rocket will be 112 feet above the ground after 1 second and after 7 seconds.

 

 

The rocket will be at its max height halfway through the flight, since this is a parabola. We know the rocket lands at 8 seconds and launches at 0 seconds, so midway through the flight is (8+0)/2 = 8/2 = 4 seconds. That means the rocket will hit max height after 4 seconds.

 

 

To find the rocket's max height, plug in t = 4 and solve for h(4):

 

h(4) = -16(4)² + 128(4)

h(4) = -16(16) + 512

h(4) = -256 + 512

h(4) = 256

 

The rocket will be at 256 at t = 4. That means that the rocket's max height is 256 feet.