Question

1. model with math an air cannon launches a t - shirt upward toward basketball fans. it reaches a maximum height and then descends for a couple seconds until a fan grabs it. sketch the graph that represents this situation.

Explanation:

Step1: Analyze the motion

The T - shirt is launched upward, so initially, as time (x - axis) increases, the height (y - axis, distance) increases. This part of the graph should be a curve (or line, but in reality, due to gravity, it's a parabola) with a positive slope that is decreasing (since the upward speed is decreasing due to gravity) until it reaches the maximum height.

Step2: Analyze the maximum height

At the maximum height, the slope of the graph (rate of change of height with respect to time) is zero. So the graph has a peak at this point.

Step3: Analyze the descent

After reaching the maximum height, the T - shirt descends. So as time increases, the height decreases. The slope of the graph in this region is negative (and its magnitude may change, but for a basic sketch, we can consider a curve with a negative slope) until the fan grabs it (the graph ends at some point on the descent, not reaching zero height necessarily, just when the fan grabs it).

To sketch the graph:

• Start at a point on the y - axis (initial height, maybe not zero if the air cannon is at some height, but we can start near the origin for simplicity) or at the origin (if we consider the launch point as height zero at time zero).
• Draw a curve that rises to a peak (maximum height) as time increases, then falls (descends) as time continues to increase, ending before reaching the x - axis (since the fan grabs it during descent). The shape is similar to the right - half of a downward - opening parabola (for the upward motion, the function is \(y = -gt^{2}+v_{0}t + h_{0}\), where \(g\) is acceleration due to gravity, \(v_{0}\) is initial velocity, and \(h_{0}\) is initial height; the graph of this quadratic function is a parabola opening downward).

Answer:

The graph should be a downward - opening parabola - like curve (or a curve with a similar shape showing increase then decrease) with the x - axis as time (in seconds) and y - axis as distance (in feet). It starts (at \(t = 0\)) with a positive slope (increasing height), reaches a maximum (where the slope is zero), and then has a negative slope (decreasing height) until the end of the graph (when the fan grabs the T - shirt). A rough sketch would have a peak (maximum height) and then a descent, with the overall shape of a "hill" (a curve that rises then falls).