A rocket is launched at t=0 seconds. It's height, in feet, above sea-level, as a function of time,
t, is given by
h(t)=-16t2+64t+192
When does the rocket hit the ground after it is launched?
-16(t^2-4t-12) =0
(t-6)(t+2) =0
t - 6 = 0
t = 6
V = Vo + g*Tr.
0 = 64 - 32Tr, Tr = 2 s. = Rise time.
h = -16Tr^2 + 64Tr + 192.
h = -16*2^2 + 64*2 + 192 = 256 Ft.
h = 0.5g*Tf^2.
256 = 16Tf^2, Tf = 4 s. = Fall time.
Tr+Tf = 2 + 4 = 6 s. To hit gnd.
To find when the rocket hits the ground, we need to find the value of t when the height, h(t), is equal to 0.
The equation for the height of the rocket is given by:
h(t) = -16t^2 + 64t + 192
Setting h(t) to 0:
0 = -16t^2 + 64t + 192
This is a quadratic equation in t. Let's solve it by factoring or using the quadratic formula.
Rearranging the equation:
16t^2 - 64t - 192 = 0
Dividing both sides by 16 to simplify the equation:
t^2 - 4t - 12 = 0
Now we need to factor this equation, or we can use the quadratic formula.
Using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 1, b = -4, and c = -12. Substituting these values:
t = (-(-4) ± √((-4)^2 - 4(1)(-12))) / (2(1))
Simplifying:
t = (4 ± √(16 + 48)) / 2
t = (4 ± √(64)) / 2
t = (4 ± 8) / 2
We have two possible values for t:
t1 = (4 + 8) / 2 = 12 / 2 = 6
t2 = (4 - 8) / 2 = -4 / 2 = -2
Since time cannot be negative, the rocket hits the ground after 6 seconds (t = 6).
To find the time when the rocket hits the ground, we need to determine the value of t for which the height, h(t), is equal to 0. We can set the equation equal to zero and solve for t as follows:
-16t^2 + 64t + 192 = 0
In order to simplify this equation, we can divide both sides by -16 to obtain:
t^2 - 4t - 12 = 0
Next, we can factor the quadratic equation if possible. In this case, the equation can be factored as:
(t - 6)(t + 2) = 0
So, either t - 6 = 0 or t + 2 = 0. Solving these equations, we find:
t = 6 or t = -2
Since time cannot be negative in this context, we can rule out the solution t = -2. Therefore, the rocket hits the ground at t = 6 seconds after it is launched.