# Calculating time until a ball fall

How exactly can I calculate time until a ball fall if I know initial and final x and y position of a ball, gravity, delta time, initial dx and dy of a ball and its speed calculated from the formula sqrt(dx ** 2 + dy ** 2)?

• Which level of precision do you need? E.g. If you want to simulate a tenis ball with rotation/effects or just a simple, ball without atmosphere? – Adrian Maire Nov 16 '16 at 12:02
• I want to simulate a small ball with no effects, without atmosphere. This equation should provide also horizontal motion, not only free fall with dx = 0 – Tkininter Nov 16 '16 at 12:23
• Downvoted as OP obviously did not make a lot of research before posting... – realUser404 Nov 16 '16 at 12:28

Supposing a macroscopic ball falling without atmosphere resistance and where Relativist effect is negligible.

• P0[x,y,z] is the initial position of the ball, with y >= r
• P[x,y,z] is the final position of the ball.
• V0[x,y,z] is the initial speed of the ball
• g[0,y,0] is the gravity, with y < 0
• r is the radius of the ball, r >= 0
• ground is defined as a plane at Y=0

The position of the ball at any moment is given by (1):

P = P0 + V0*t + g*t*t;


To found the time where the ball hit the ground (2):

P0.y + V0.y*t + g.y*t*t = r


Solving the equation (3):

t = ( -V0.y - sqrt(V0.y*V0.y - 4*g.y*(P0.y-r)) ) / (2 * g.y)


Final position can be calculated using (1)

Note: Considering that P0.y >=r and g.y is negative, the internal of the root is always positive.

an unusual answer cause its mostly a link, but this is the way to calculate object falling according to X gravity following newton's law.

https://en.wikipedia.org/wiki/Equations_for_a_falling_body