why are these equations poor models of hitting a baseball:
y= -0.002x(x-433.1)
I get maximums of (216,93). Other problems' maximums are in the (220, 100)range. So are they poor models because the distance and heights are unreasonable?
2) Why is the constant a in y=ax^2 + bx + c negative in a reasonable model?
is it because the graph is then a hill versus a valley?
1. Plenty of players hit the ball 400 feet or more. This function has roots at 0,433, so it appears to model typical hitting ability.
2. negative a means that there is a downward force acting on the ball: gravity. so, yes, the graph is a hill. The ball takes off at some speed, but gravity slows down its ascent and makes it drop back to earth.
y= -0.002x(x-433.1)
is
y = .8662 x - .002 x^2
I do not see anything very wrong except that the baseball does not start out at zero height. When x = 0 y should be like one meter high so I might prefer something like
y = 1 + .8662 x - .002 x^2
Yes, coef of x^2 must be negative because as x gets big the ball must drop.
These equations might be considered poor models of hitting a baseball for a couple of reasons:
1) Unreasonable Distance and Height: In the first equation, y = -0.002x(x-433.1), the maximum values you obtained for distance and height are (216, 93). Comparing this to other models that have maximums in the range of (220, 100), it suggests that the predicted distances and heights in this equation are not aligned with realistic expectations. Hitting a baseball with a maximum height of 93 feet or a maximum distance of 216 feet seems highly unlikely.
2) Negative Constant 'a' in Quadratic Equation: In the second equation, y = ax^2 + bx + c, if the constant 'a' is negative in a reasonable model, it implies that the graph would represent a downward-opening parabola or a hill-like shape rather than an upward-opening parabola or a valley-like shape. This downward-opening shape doesn't align with the expected trajectory of a baseball hit in the air, where the ball typically follows a rising path after being struck.
When creating models for hitting a baseball, it is important to consider realistic physical constraints, such as the height and distance the ball can reach, as well as the expected shape of the trajectory.