Toy Rocket Problem

Question

A toy rocket is launched from the ground level with an initial vertical velocity of 96 ft/s. After how many seconds will the rocket hit the ground?

Answer 1

Projectiles subject to gravity can be modeled by the following equation:

 

h(t) = (1/2)at² + v₀t + h₀,

 

where h(t) is the object's height, a is acceleration due to gravity, t is time that has passed, v₀ is initial velocity, and h₀ is initial height.

 

 

 

You're given an initial velocity of v₀ = 96 ft/s, and you're told the rocket is launched from ground level, so initial height h₀ = 0 ft. Acceleration due to gravity is approximately a = -32 ft/s². Let's plug those values into the equation:

 

h(t) = (1/2)(-32)t² + 96t + 0

h(t) = -16t² + 96t

 

 

 

Finally, you're asked how many seconds it will take for the rocket to hit the ground. That means we'd like to know what the time t is when the height h(t) = 0. So, plug in h(t) = 0 and solve for t by factoring:

 

h(t) = -16t² + 96t

0 = -16t² + 96t

0 = -16t(t - 6)

0 = -16t    OR    0 = t - 6

0 = t    OR    6 = t

 

The rocket is on the ground at t = 0 seconds (when it is launched) and at t = 6 seconds (when it lands). We're looking for the time that passes before the rocket lands, so that's t = 6 seconds.