While the ball falls, gravity speeds it up at a constant rate. When acceleration is constant, position changes with time according to this equation:
y = y₀ + v₀t + ½at²
The app finds the curve y = At² + Bt + C that comes closest to all of your points. That's the line drawn on the graph. It picks A, B, and C so that the vertical gaps between the curve and your points, squared and added up, are as small as possible.
Matching the two equations term by term:
- C matches y₀, the starting position. It comes out near 0 because your first mark is the origin.
- B matches v₀, the ball's velocity at your first mark. It isn't 0 because you started marking after the ball was already falling.
- A matches ½a. So the acceleration is a = 2A.
A thrown ball's motion can be split into a horizontal part and a vertical part. The two happen at the same time without affecting each other.
Horizontal. After the ball leaves the hand, nothing pushes it sideways (ignoring air resistance), so it moves sideways at a steady speed:
x = x₀ + vₓt
The app fits a straight line x = Dt + E to your horizontal positions. The slope D is the horizontal velocity, vₓ.
Vertical. Gravity speeds the ball up downward at a constant rate, exactly as it does for a dropped ball:
y = y₀ + v₀t + ½at²
The app fits the curve y = At² + Bt + C, choosing A, B, and C so that the vertical gaps between the curve and your points, squared and added up, are as small as possible. Matching the two equations term by term:
- C matches y₀, the starting position, near 0 because your first mark is the origin.
- B matches v₀, the ball's vertical velocity at your first mark. Positive means it was still rising.
- A matches ½a. So the acceleration is a = 2A.