A baseball is thrown straight up in the air with an initial velocity of 29 feet per second from a point exactly 6 feet off the ground.

Question

i need to do this as a math project and i need help with it. i need to state the window, the xmin and max, the y min and max, and heres the questions i also need answered: After how much time in the air will the baseball land on the ground? Eleanora catches the ball on its way back down when it is 5 feet off of the ground. How much time has passed after it is thrown before she catches the ball? When will the ball reach its maximum height? What will that maximum height be? How high above the ground will the baseball be 0.7 seconds after it is launched?

If someone could help me with this, id really appreciate it. (you also are probably gonna need a ti84 graphing calculator)

catches the ball on its way back down when it is 5 feet off of the ground. How

much time has passed after it is thrown before she catches the ball?

After how much time in the air will the baseball land on the ground?

Answer 1

final velocity = initial velocity + acceleration x time

And maximum height is when the final velocity is 0

0 = 29 + (-32)t

-32t = -29

t = 0.9 seconds approximately until it reaches max height (extra digits are insignificant)

total height(y) = initial distance + initial velocity x time + 1/2acceleration x time^2

y = 6 + 29t + 0.5(-32)t^2

y = -16t^2 + 29t + 6

y = -16(0.9)^2 + 29(0.9) + 6 = 19ft at maximum height approximately

Total distance when the ball hits the ground is 19 + 19 - 6 (19 - 6 feet up into the air + 19 feet down)

= 32 feet

Height at 0.7 seconds:

y = -16(0.7)^2 + 29(0.7) + 6 = 24ft

For the graph, assuming x is time and y is distance:

Y should be 0 to 20 (feet) (because max height was 19ish)

X should be 0 to 2 (seconds) (because max height was 0.9 seconds to go 13 feet and it will go 14 feet until caught)

***Mathematically:

The equation will look like y = -16x^2 +29x + 6 (from the kinematic equation 0.5at^2 + vt + starting distance)

(Also, derived with calculus: Gravity acceleration = -32 feet per second^2

The integral of this is -32x which is the speed (the acceleration of -32 is the rate of change or slope of this)

The integral of -32x is -16x^2 from which we can derive our function)

To solve for total time: set this function equal to 0 (when "y" or distance returns to 0 feet):

-16x^2 +29x + 6 = 0

-(x - 2)(16x + 3) = 0

x = 2 seconds (the other value is negative and does not exist)