Using the formula
h=-16^2+vt+s
Answer the question: A rocket is shot upward off a 25 foot high balcony with an initial velocity of 96 feet per second. How many seconds will it take until the rocket hits the ground?
I think the formula should read:
h=-16t^2+vt+s
Where s=initial height = 25'
v=initial velocity = 96 fps
t=time in seconds
-16=half of acceleration due to gravity in ft/s²
so the equation becomes
h(t)=-16t²+96t+25
solve for t when h(t)=0.
To find the time it will take for the rocket to hit the ground, we can use the formula h = -16t^2 + vt + s, where:
- h is the height (in this case, it will be 0, since the rocket will hit the ground)
- t is the time it takes for the rocket to hit the ground
- v is the initial velocity (given as 96 feet per second)
- s is the initial height (given as 25 feet)
Substituting the given values into the formula, we have:
0 = -16t^2 + 96t + 25
To solve this quadratic equation for t, we can either factor it or use the quadratic formula. In this case, it is easier to use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2a
Plugging in the values for a, b, and c from the equation above, we get:
t = (-96 ± √(96^2 - 4(-16)(25))) / (2(-16))
Simplifying further:
t = (-96 ± √(9216 + 1600)) / (-32)
t = (-96 ± √(10816)) / (-32)
t = (-96 ± 104) / (-32)
Now we can solve for the two possible values of t:
t1 = (-96 + 104) / (-32) = 8 / (-32) = -0.25
t2 = (-96 - 104) / (-32) = -200 / (-32) = 6.25
Since time cannot be negative, the rocket will hit the ground after approximately 6.25 seconds.
To find the number of seconds it takes for the rocket to hit the ground, we need to find the value of "h" when it equals 0.
The equation provided is:
h = -16t^2 + vt + s
Given that the rocket is shot upward off a 25-foot high balcony, the initial height (s) is 25 feet. The initial velocity (v) is 96 feet per second, and we need to find the time (t) when the height (h) equals 0.
Plugging in the given values into the equation:
0 = -16t^2 + 96t + 25
Now, we have a quadratic equation. To solve for "t," we can either factorize, complete the square, or use the quadratic formula.
Let's use the quadratic formula:
Given an equation in the form of ax^2 + bx + c = 0, the quadratic formula states:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, our equation is:
-16t^2 + 96t + 25 = 0
Comparing this to the quadratic form ax^2 + bx + c = 0, we have:
a = -16, b = 96, c = 25
Plugging these values into the quadratic formula, we get:
t = (-96 ± √(96^2 - 4(-16)(25)) / (2(-16))
Simplifying the equation further:
t = (-96 ± √(9216 + 1600) / -32
t = (-96 ± √10816) / -32
t = (-96 ± 104) / -32
Now, we have two possible solutions for t:
t = (-96 + 104) / -32 = 8 / -32 = -0.25 seconds
t = (-96 - 104) / -32 = -200 / -32 = 6.25 seconds
Since we are looking for the time it takes for the rocket to hit the ground, we discard the negative value (-0.25 seconds). Therefore, the rocket will hit the ground approximately 6.25 seconds after being shot upward.