determine quadratic function of a baseball....and find maximum height of the baseball.?

Question

The distance from Homeplate to the leftfield wall is 418.75 feet. baseball player hits the ball 3 feet above homeplate. PeeWee Reese is playing shortstop and he jumps and reaches 10 feet into the air but the ball clears his mit by a few inches. The ball travels out towards leftfield and goes over the wall by a foot and is a homerun. The left field wall is exactly 6 feet. The height of the ball varies quadratically with the distance from homeplate. a Baseball Diamond is a square with the sides 90 feet. Find a quadratic function describing the ball's homerun and determine the maximum height of the baseball.

Answer 1

A typical motion problem in two dimensions; I answer these things all day over in the Physics section :)

The origin of the coordinate system is at home plate. The +x axis is towards PeeWee and the left-field wall, and the +y axis is straight up.

The vertical motion of the ball is found using the Galilean Equations of Motion:

Vertical Dimension:

y - yo = voy t - 1/2 g t^2

where y is the final vertical displacement (3 feet), yo the initial vertical displacement (7 feet), voy the initial vertical velocity (unknown), t the elapsed time, and g the local acceleration of gravity (-32 feet/s^2). This is a quadratic equation in time.

Horizontal Dimension:

x - xo = vox t

where x is the final horizontal position (418.75 feet), xo the initial horizontal position (zero), vox the initial horizontal velocity (unknown), and t the elapsed time (same t as in the other dimension).

Rewrite the second equation as t = (x - xo) / vox and substitute into the first. You've now eliminated time and the remaining unknowns are vox and voy. Note that vox^2 + voy^2 = v^2 (Pythagorean Theorem) and that voy / vox is the tangent of the angle at which the ball was initially hit.

Answer 2

for y = ax^2 + bx +c then if a is effective, the function has a minimum. If a is damaging, the function has a maximum. Differentiate the function. You get 2ax+b. The minimum or maximum is the place the dervative is 0 because of the fact this is the place the slope is 0. this is because of the fact if the slope isn't 0, then moving to the left or the suitable alongside the curve could bring about a extra robust or decrease fee, so the standards the place the slope isn't 0 can't be maximums or minimums. remedy for x whilst the by-product 2ax+b is 0. You get x=-b/2a. this is the x place the place the minimum or maximum takes place. Plug x=-b/2a into the unique equation and you get the fee of y on the min or max, this is the minimum or maximum of the function.