Question

part g
based on your observation of the tracker videos and the calculations of average vertical velocity and acceleration,

• why do you think the vertical acceleration values are not equal to the theoretical acceleration due to gravity, -9.81 meters/second²?
• what can you say about your earlier prediction about which ball will land first? if your hypothesis was wrong, can you propose an alternate rationale for the observed results?

Explanation:

For the first question (Why vertical acceleration ≠ -9.81 m/s²):

1. Air Resistance: The theoretical value (-9.81 m/s²) assumes a vacuum. In reality, the ball experiences air resistance, which opposes motion and reduces the net downward acceleration (since \( F_{\text{net}} = F_g - F_{\text{air}} \), so \( a=\frac{F_{\text{net}}}{m}=\ g-\frac{F_{\text{air}}}{m} \), making \( |a| < 9.81 \, \text{m/s}^2 \)).
2. Experimental Errors: Tracker video analysis may have errors:

• Position Tracking: Inaccurate marking of the ball’s position in frames (e.g., blurry video, misaligned markers) introduces errors in calculated velocity/acceleration.
• Frame Rate: If the video’s frame rate is not precisely known, time intervals (\( \Delta t \)) for velocity calculation (\( v=\frac{\Delta y}{\Delta t} \)) and acceleration (\( a=\frac{\Delta v}{\Delta t} \)) are miscalculated.
• Calibration: The video’s scale (converting pixel distance to real - world meters) may be incorrectly calibrated, distorting displacement values.

For the second question (Prediction about which ball lands first):

Case 1: Prediction was correct

If your hypothesis (e.g., “Both balls land simultaneously” for a free - fall vs. projectile with same initial vertical conditions) matched, it confirms that vertical motion (governed by \( y = y_0+v_{0y}t-\frac{1}{2}gt^2 \)) is independent of horizontal motion (for projectiles). The time to land depends only on initial vertical position (\( y_0 \)), initial vertical velocity (\( v_{0y} \)), and gravity—horizontal velocity (\( v_{0x} \)) does not affect the time to fall.

Case 2: Prediction was wrong

If you predicted, say, “The horizontally launched ball lands later” (wrong), the alternate rationale is the same as above: vertical motion is independent of horizontal motion. For two balls with the same \( y_0 \) and \( v_{0y} = 0 \) (or same \( v_{0y} \)), their vertical displacement equations (\( y - y_0=v_{0y}t-\frac{1}{2}gt^2 \)) are identical. So time to land (\( t \)) depends only on \( y_0 \), \( v_{0y} \), and \( g \)—not \( v_{0x} \). If the error was in assuming horizontal motion affects fall time, this corrects that misunderstanding.

Answer:

s:

1. Vertical acceleration differs from \(-9.81\ \text{m/s}^2\) due to air resistance (opposing motion, reducing net acceleration) and experimental errors (position tracking, frame rate, calibration).
2. If prediction was correct: Confirms vertical motion is independent of horizontal motion. If wrong: Vertical motion (fall time) depends on \( y_0 \), \( v_{0y} \), \( g \)—not horizontal velocity (\( v_{0x} \)), so balls with same vertical conditions land simultaneously.