Question

part 1: air table
objective: use measurements of position over time to calculate the horizontal and vertical motion of a projectile.
do you predict that the horizontal velocity will increase, decrease, or stay constant over time?
do you predict that the vertical velocity will increase, decrease, or stay constant over time?
data and analysis
find the horizontal velocity, in terms of \boxes/tick.\ show your calculation or reasoning here. what is the horizontal acceleration?
find the vertical acceleration, in terms of \boxes/tick².\ show your calculation or reasoning here.
pick 8 - 10 points from the paper. on a separate sheet of graph paper, draw four graphs, for the horizontal position and horizontal velocity over time, and the vertical position and vertical velocity over time. staple it here.
do the horizontal motion graphs exhibit constant velocity or acceleration? explain, using the graphs as evidence.
do the vertical motion graphs exhibit constant velocity or acceleration? explain, using the graphs as evidence.

Explanation:

Step1: Predict horizontal velocity

In the absence of horizontal - forces (assuming no air - resistance etc. on the air - table), the horizontal velocity of a projectile stays constant. This is based on Newton's first law of motion.

Step2: Predict vertical velocity

There is a constant downward acceleration due to gravity acting on the projectile. So, the vertical velocity of a projectile increases over time.

Step3: Calculate horizontal velocity

Let \(x_1\) and \(x_2\) be the horizontal positions at times \(t_1\) and \(t_2\) (where \(t_2 - t_1=\Delta t\) is the time between two consecutive ticks). If the horizontal positions are measured in terms of boxes, the horizontal velocity \(v_x=\frac{x_2 - x_1}{t_2 - t_1}\) (in boxes/tick). Since there is no horizontal force, the horizontal acceleration \(a_x = 0\) (from Newton's second law \(F = ma\), and \(F_x=0\)).

Step4: Calculate vertical acceleration

Let \(y_1\), \(y_2\), \(y_3\) be the vertical positions at times \(t_1\), \(t_2\), \(t_3\) (\(t_2 - t_1=t_3 - t_2=\Delta t\)). The vertical acceleration \(a_y=\frac{(y_3 - y_2)-(y_2 - y_1)}{\Delta t^2}\) (in boxes/tick²).

Step5: Analyze horizontal motion graphs

If the horizontal velocity is constant, the graph of horizontal position \(x\) vs time \(t\) will be a straight - line with a non - zero slope (since \(x = v_x t\)), and the graph of horizontal velocity \(v_x\) vs time \(t\) will be a horizontal line at \(v_x\) (because \(v_x\) is constant).

Step6: Analyze vertical motion graphs

The graph of vertical position \(y\) vs time \(t\) will be a parabola (\(y=v_{0y}t+\frac{1}{2}a_y t^2\)) and the graph of vertical velocity \(v_y\) vs time \(t\) will be a straight line with a non - zero slope (since \(v_y=v_{0y}+a_y t\)), indicating constant acceleration.

Answer:

Horizontal velocity prediction: Stays constant.
Vertical velocity prediction: Increases.
Horizontal velocity calculation: \(v_x=\frac{x_2 - x_1}{t_2 - t_1}\) (boxes/tick), horizontal acceleration \(a_x = 0\).
Vertical acceleration calculation: \(a_y=\frac{(y_3 - y_2)-(y_2 - y_1)}{\Delta t^2}\) (boxes/tick²).
Horizontal motion graphs: Constant velocity (straight - line for \(x\) vs \(t\) and horizontal line for \(v_x\) vs \(t\)).
Vertical motion graphs: Constant acceleration (parabola for \(y\) vs \(t\) and straight line with non - zero slope for \(v_y\) vs \(t\)).