An explosion causes debris to rise vertically with an initial velocity of 160 feet per second.
In how many seconds does it attain maximum height?
i know i need to use the formula:
y(t)=-16t^2+Vot + yo
Have:
vo=160
Yo= is what i am trying to find
and i dont know how to solve for time
so you know:
height = -16t^2 + 160t + 0 , zero, since we started at ground level
This is a downwards opening parabola, so it has a maximum at its vertex.
What method have you learned to find the vertex, there are several ways.
simplest way in this case:
find the x-intercepts, then find the midpoint
h = -16t^2 + 160t
= -16t(t - 10)
so the intercepts are 0 and 10,
then half-way time is 5 seconds
Well, solving for time can sometimes be tricky. But don't worry, I'm here to help!
Let's start by finding the maximum height. We know that at maximum height, the vertical velocity (Vy) is zero. That means we need to find the time when Vy equals zero.
Using the formula for vertical velocity, Vy = Vo - 32t, we can set Vy equal to zero and solve for t:
0 = 160 - 32t
32t = 160
t = 160/32
t = 5
Great! So it takes 5 seconds for the debris to reach maximum height. But we're not done yet! We still need to find the initial height (Yo).
Now, we know that at the maximum height, the final vertical velocity (Vy) is also zero. Using the formula for vertical displacement, we have:
0 = -16t^2 + 160t + Yo
We can substitute t = 5 into this equation and solve for Yo:
0 = -16*(5^2) + 160*5 + Yo
0 = -16*25 + 800 + Yo
0 = -400 + 800 + Yo
0 = 400 + Yo
Yo = -400
So, the debris was initially 400 feet below the ground level. I hope that clears things up!
To solve for the time it takes for the debris to reach its maximum height, we need to find the time when the vertical velocity (Vy) becomes zero.
In this case, the initial vertical velocity (Vo) is 160 feet per second, and the acceleration due to gravity (g) is -32 feet per second squared (considered negative because it is acting downward).
y(t) = -16t^2 + Vot + yo
Since we are trying to find the time it takes for the debris to reach maximum height, we know that at that point, the vertical velocity is zero. So, Vy = 0.
Vy = Vot + g * t
Since Vy is zero, we can substitute the values:
0 = 160 - 32t
Solving for t:
32t = 160
t = 160 / 32
t = 5
Therefore, it takes 5 seconds for the debris to reach its maximum height.
To solve for the time it takes for the debris to attain maximum height, we need to find the value of t for which the velocity becomes zero.
The formula for vertical velocity as a function of time is given by:
v(t) = vo - 32t,
where vo is the initial velocity and the term -32t represents the acceleration due to gravity.
In this case, we have vo = 160 ft/s, and we want to find the value of t when v(t) = 0.
Setting v(t) = 0:
0 = 160 - 32t.
Now, solve for t:
32t = 160,
t = 160 / 32,
t = 5 seconds.
Therefore, it takes 5 seconds for the debris to reach maximum height.