Question

Jean throws a ball with an initial velocity of 64 feet per second from a height of 3 feet. Write an equation and answer the questions below. Show all your work for full credit. Use the correct units with your answers

Answer 1

Answer:

Step-by-step explanation:

From what I understand about parabolic motion in the English system, the equation for flight is

$s(t)=-16t^2+v_{0}t+h_{0}$

where $v_{0}t$ is the initial upwards velocity and

$h_{0}$ is the initial height from which the object was launched.  Filling in that equation with those values gives you

$s(t)=-16t^2+64t+3$.  That's a.

In order to determine how long it will take the ball (or rocket...the problem is mixing up the 2) to reach its max height you need to put the equation into vertex form, since the vertex of a parabola is the absolute max (or min depending upon the parabola) of the function.  The absolute max is the heighest that the ball will go.  Completing the square is the way to solve this.  Begin by setting the equation equal to 0, the moving the 3 over by subtraction:

$-16t^2+64t=-3$

Now factor out the -16 since the leading coefficient HAS to be a positive 1:

$-16(t^2-4t)=-3$

Now take half the linear term (half of 4t which is 2), square it (4) and add it into the parenthesis:

$-16(t^2-4t+4)=-3$

BUT since you added in a 4*-16 on the left you have to add it in on the right:

$-16t^2(t^2-4t+4)=-3-64$

which simplifies to

$-16(t-2)^2=-67$

Now bring the 67 over by addition and you have your vertex:

$s(t)=-16(t-2)^2+67$.

The vertex is (2, 67).  The 2 stands for time, so 2 seconds, and the 67 stands for feet, so at 2 seconds the max height is 67 feet.

How long it will be in the air is found by factoring to find the zeros.  These can be found by plugging the quadratic into the quadratic formula and getting that the zeros are -0.046 and 4.046

So the quadratic starts a tiny tiny bit to the left of the origin, but for all intents and purposes we can say it starts at the origin (x = 0) and ends at

x = 4.05 seconds.  Which makes sense if you know anything about parabolic motion and physics.  The vertex indicates not only the time and the max height at that time, it also is indicative of the halfway mark.  Meaning that if it takes 2 seconds to reach its max height, it will hit the ground at 4 seconds.  And 4.05 is close enought to 4 (but since you were told to round to the nearest hundredth, that .05 matters).  Sorry it's so long, but it's not a question that can be answered with just a few sentences.