Vernon Wells hits a baseball that travels for 142m before it lands. The flight of the ball can be modeled by a quadratic function in which x is the horizontal distance the ball has traveled away from Vernon, and h(x) is the height of the ball at that distance.
Assume that the ball was between 0.6m and 1.5m above the ground when it was hit:
a)What would be a good range of values for the height of the ball? are some values for the height unreasonable?
b)What happens when x=0?
c) What are the possible values for h(x) when x=0?
d)What would be a good range of values for the height of the ball? Are some values for the height unreasonable?
(b) h(0) is the height at time 0: when the ball was hit
(c) so, 0.6 <= h(0) <= 1.5
(a) Since most balls are hit at a fairly low trajectory, I'd say that since the max height of the ball occurs at roughly 71 meters, that might be a good estimate for the height of the ball at that point.
(d) is exactly the same as (a). The maximum height is
(v sinθ)^2/(2g)
maybe you should investigate the speed at which baseballs are usually hit. Then since θ=45° gives the maximum possible height, use your value for v to get a greatest possible height of v^2/(4g).
a) Since the ball was hit between 0.6m and 1.5m above the ground, a reasonable range for the height of the ball would be between 0.6m and 1.5m. Values outside this range would be considered unreasonable as they would not reflect the initial conditions provided.
b) When x=0, it means the ball is at the starting point, which is the position of Vernon when he hit the ball.
c) To find the possible values for h(x) when x=0, we need to substitute x=0 into the quadratic function representing the flight of the ball. Since we do not have the specific equation for the quadratic function, we cannot determine the exact values without further information.
d) A good range of values for the height of the ball would be between 0.6m and 1.5m, as mentioned earlier. Some values for the height that would be considered unreasonable could be negative values or heights greater than 1.5m, as they would not align with the given initial conditions.
To determine a good range of values for the height of the ball, let's consider the given information. We know that the ball traveled for 142m before it landed, which means the vertex of the quadratic function representing the ball's flight is located at the halfway point.
The vertex form of the quadratic function is given by h(x) = a(x - h)^2 + k, where (h, k) represents the coordinates of the vertex. In this case, since the vertex is at the halfway point, h would be half of 142m, which is 71m.
Now, let's analyze the given height range of the ball, which is between 0.6m and 1.5m above the ground. Since the vertex represents the maximum/minimum height of the ball, the range of values for a good height of the ball would be h(71) ± (1.5 - 0.6) meters. This gives us a range of values between h(71) + 0.9 and h(71) - 0.9 meters.
To determine whether some values for the height are unreasonable, we need to consider the physics of the scenario. Normally, a baseball player hits the ball with a relatively low angle, and the height is usually fairly close to the ground. Values significantly higher than 1.5m may be considered unreasonable, as it suggests an abnormally high shot. Conversely, values lower than 0.6m may also be considered unrealistic for a baseball hit.
Now let's address each of the specific questions:
a) The range of values for the height of the ball would be h(71) ± 0.9 meters. Values significantly higher than 1.5m or lower than 0.6m may be considered unreasonable.
b) When x = 0, it means the ball is right at the starting point, which is where Vernon Wells hit the ball. At x = 0, we have h(0), which represents the height of the ball when it was hit.
c) The possible values for h(x) when x = 0 would be h(0), which is the height of the ball when it was hit. We would need additional information or the specific quadratic function to calculate this value.
d) As mentioned earlier, a good range of values for the height of the ball would be h(71) ± 0.9 meters. Values significantly higher than 1.5m or lower than 0.6m may be considered unrealistic in the context of a baseball hit.