Visualisation

Calculation

Let $A$ be the coordinates of the first ball and $B$ the coordinates of the second ball:

$$ A = (x_a, y_a) $$

$$ B = (x_b, y_b) $$

Let’s define their respective velocity vectors:

$$ v_a = (x_{va}, y_{va}) $$

$$ v_b = (x_{vb}, y_{vb}) $$

Determine the threshold distance between the two balls to detect a contact:

$$ \sqrt{(x_b-x_a)^2+ (y_b-y_a)^2} \leq 2r $$

where $r$ is the ball radius.

At the point of contact, calculate the vector between the centre points of both balls:

$$ v_{ab} = (x_b-x_a, y_b-y_a) $$

This vector has the direction in which ball A will continue rolling after the impact.

To determine the ball velocities after the impact, we follow the conservation of momentum:

$$ v_a = v^{'}_a + v^{'}_b $$

and conservation of energy:

$$ {|v|_a}^2 = {|v|'_a}^2 + {|v|'_b}^2 $$

This can only be true if the angle between $v^{'}_a$ and $v^{'}_b$ is 90 degrees.

The magnitudes are calculated as follows:

$$ |v|'_a = |v|_a \times cos{\phi} $$

$$ |v|'_b = |v|_b \times sin{\phi} $$

Resources

• 90 degree rule
• Momentum - Multiple dimensions

Visualise lines pointing in the direction of the velocity of each ball on the table