Question

part c
now observe the graph of the vertical velocity against time ($v_y$ vs. $t$). to do so, click the “$y$” label on the vertical axis and select “$v_y$: velocity y - component” from the pop - up menu.
what do you observe about the vertical velocity of the ball? what does this say about the mathematical relationship between y - velocity

Explanation:

When observing the vertical velocity (\(v_y\)) vs. time (\(t\)) graph for a ball (typically in projectile motion or free - fall with gravity), the vertical velocity of the ball changes linearly with time. This is because the only significant vertical force acting on the ball (neglecting air resistance) is gravity, which gives a constant acceleration (\(a = -g\) for downward as negative, or \(a = g\) for downward as positive) in the vertical direction. According to the kinematic equation \(v_y=v_{y0}+at\) (where \(v_{y0}\) is the initial vertical velocity and \(a\) is the acceleration), since \(a\) is constant, \(v_y\) is a linear function of \(t\). So the graph of \(v_y\) vs. \(t\) is a straight line, indicating a linear mathematical relationship between vertical velocity and time (the velocity increases or decreases at a constant rate, i.e., with a constant slope which is equal to the acceleration due to gravity).

Answer:

When observing the \(v_y\) vs. \(t\) graph, the vertical velocity of the ball changes linearly with time (the graph is a straight line). This shows that there is a linear mathematical relationship between \(y\) - velocity and time, as described by the kinematic equation \(v_y = v_{y0}+at\) (where \(a\) is the constant acceleration due to gravity), meaning velocity changes at a constant rate (constant slope on the \(v_y - t\) graph).