If an object is launched straight up into the air from a starting height of h_{0} feet, then the height of the object after t seconds is approximately h=-16t^2+v_{0}t+h_{0} feet, where v_{0} is the initial velocity of the object. Find the starting height and initial velocity of an object that attains a maximum height of 412 feet five seconds after being launched.
Did you not look how your previous two posts of the same question came out?
Hard to guess what you mean by
h_{0} and v_{0}t
That is why your question does not get answered
Knowing the formula, it seems clear that h_{0} means h0.
So, forgetting the subscripts, let's just say
y = -16t^2 + vt + h
y' = -32t + v
y'=0 when t=5
-32(5) + v = 0
v = 160
y = -16t^2 + 160t + h
412 = -16(25)+160(5)+h
h = 12
To find the starting height and initial velocity of the object, we can start by analyzing the given information.
We know that the object reaches its maximum height after five seconds. At this point, the velocity of the object will be zero because it momentarily stops before falling back down. Therefore, we can set up the equation:
0 = -16(5)^2 + v_0(5) + h_0
Simplifying further, we have:
0 = -16(25) + 5v_0 + h_0
0 = -400 + 5v_0 + h_0
Since the maximum height of the object is given as 412 feet, we can also set up another equation:
h = -16(5)^2 + v_0(5) + h_0 = 412
Substituting the values, we have:
-400 + 5v_0 + h_0 = 412
Now, we have a system of two equations with two variables (v_0 and h_0). We can solve this system using substitution or elimination.
Let's use substitution to solve for one variable in terms of the other.
From the first equation, we can rewrite h_0 in terms of v_0:
h_0 = 400 - 5v_0
Now substitute this expression for h_0 in the second equation:
-400 + 5v_0 + (400 - 5v_0) = 412
Simplifying further:
-400 + 5v_0 + 400 - 5v_0 = 412
0 = 12
The result is an equation that is not true, which means there is no solution for this system of equations. Therefore, it is not possible to determine the starting height and initial velocity of an object that attains a maximum height of 412 feet five seconds after being launched with the given information.