Find the equation of a ball and it's height.

Question

A student estimated based on the video that the ball left my hand 1.59 meters off the ground, and after 0.58 seconds the ball reached the maximum height of 3.33 meters.

Use this information to find an equation of the form h=a(t-t1)^2 + h1

I am able to find the answer to everything behind the equation, but cannot find the answer to the h. An example of a previous problem's answer was -4.96984(t+-0.39)^2+3.33. I am unsure of how to get the h number.

Answer 1

Given:

The vertex of the equation is the point at which the maximum or minimum is attained. For degree 2 (x²), there exists 1 vertex. Because the ball is rising initially, then falling beyond the peak, the vertex will be concave downward, implying that the equation is of the form (Ax²) + h1 where A < 0 and x = (t-t1).

For this case, the vertex is attained where the new origin is defined (creating symmetry on both sides of vertex). The origin is defined such that (x) = 0 or (t-t1) = 0. The vertex is attained at the peak of the function, which occurs when t = 0.58 seconds--> (0.58 - t1) = 0 or t1 = 0.58.

Also, every point before or beyond the vertex must be less than the maximum (function shaped concave down). This implies that Ax² + h1 = 3.33 when x = (t-0.58) = 0 --> h1 = 3.33.

We know the ball is initially at a height of 1.59 m at t = 0--> (t-t1) = (0-0.58) -0.58.

Therefore, Ax² +3.33 = 1.59. Substituting for x then yields.... A (-0.58)² +3.33 = 1.59 -->.3364*A + 3.33 = 1.59.

Subtracting 3.33 from both sides yields... .3364*A = -1.74 (the ball traveled a negative distance of -1.74 m from t = 0.58 back to t=0).

Solving for A yields....-1.74/.3364 = -5.17 = A.

So, the equation becomes -5.17(t-t1)² +3.33

......................................................................

Check:

When t = 0, (t-t1) = (0-.58) = -.58 and -5.17*(-.58)² +3.33 = -5.17*.3364 + 3.33 = 1.59 m.

Answer 2

Equation of motion of ball h(t) = a(t-0.58)² + 3.33.

h(0) = a(0-0.58)² +3.33 = 1.59, a ≅ - 5.17m/s²_.

_{Because h(t) = - 5.17(t - 058)}² _{+ 3.33}