A projectile is launched from a cannon that sits 4 feet off the ground at an initial velocity of 120 ft/sec. Find the position function and how long is it before the projectile hits the ground?
I got -16t^2 +120t+4 for my position function, but I'm not sure how to find the other part?
Also, how do I find the maximum height?
let h(t) be the position function
h(t) = - 16t^2 + 120t + 4 . you had that.
when it hits the ground, h(t) = 0
-16t^2 + 120t + 4 = 0
4t^2 - 30t -1 = 0
solve with quadratic formula
t = (30 ± √916)/8 = 7.533 or a negative value of t
it will hit the ground after appr 7.5 seconds
h(t) is a parabola opening downwards, so it has a maximum at its vertex.
I don't know what method you have learned to find the vertex, a common method is to complete the square.
OR
the x of the vertex is -b/(2a) = -120/-32 = 15/4 or 3.75
h(15/4) = -16(225/16) + 120(15/4+ + 4
= 229 ft
To find the position function, let's assume that the projectile is launched at time t=0.
The position function of the projectile is given by the equation:
s(t) = -16t^2 + v0t + s0
In this equation, s(t) represents the position of the projectile at time t, v0 represents the initial velocity, and s0 represents the initial position (in this case, the height of the cannon).
Given that the initial velocity is 120 ft/sec and the initial position is 4 feet, the position function becomes:
s(t) = -16t^2 + 120t + 4
Now, to find out how long it takes for the projectile to hit the ground, we need to find the value of t when the position, s(t), is equal to 0. In other words, we need to solve the equation -16t^2 + 120t + 4 = 0.
To solve this quadratic equation, you can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = -16, b = 120, and c = 4. Substituting these values into the quadratic formula, we get:
t = (-120 ± √(120^2 - 4(-16)(4))) / (2(-16))
Simplifying further, we get two values for t:
t = (-120 ± √(14400 + 256)) / (-32)
Calculating the values, we have:
t1 = (-120 + √(14400 + 256)) / (-32)
t2 = (-120 - √(14400 + 256)) / (-32)
Now, we can determine the positive value of t since time cannot be negative in this context. So, substitute the values into the equation and calculate to find out the time it takes for the projectile to hit the ground.