Two-dimensional collisions

In the real world, many collisions occur in two dimensions.

To analyse these:

• Break down the momentum vectors into x and y components.
• Apply the conservation of momentum separately for each component.

This approach allows for a more detailed analysis of collisions that occur in multiple directions, helping to understand the dynamics of each object involved.

Worked example: Analysing a two-dimensional collision

Two hockey pucks collide on a frictionless surface. Puck A (mass 0.5 kg) moves at 4 m s^-1^ in the positive x-direction. Puck B (mass 0.3 kg) moves at 3 m s^-1^ in the positive y-direction. After the pucks collide, puck A moves at 3.33 m s^-1^ at an angle of 58.5° to the horizontal. Calculate the speed and direction of puck B after the collision.

Step 1: FormulaConservation of momentum in two dimensions:

• ${\text{Momentum}}_{\text{initial x}}={\text{Momentum}}_{\text{final x}}$
• ${\text{Momentum}}_{\text{initial y}}={\text{Momentum}}_{\text{final y}}$

Step 2: Calculate Initial MomentaMomentum = mass x velocity

• ${\text{Momentum}}_{\text{A initial x}}$ $=0.5\times 4=2{\text{kg m s}}^{-1}\text{ (in x-direction)}$
• Momentum_{B initial y}​ = 0.3 × 3 =0.9 kg m s^-1 (in y-direction)

Step 3: Calculate the x and y components of puck A after the collision

v_x = 3.33 cos (58.5) = 1.73992 m s^-1^

v_y = 3.33 sin (58.5) = 2.83929 m s^-1^

Step 4: Apply Conservation of MomentumIn the x-direction:

2 = (0.5 x 1.7399) + 0.3 x v_{B x direction}

In the y direction:

0.9 = (0.5 x 2.83929) + 0.3 x v_{B y direction}

Step 5: Solve for final x and y components of puck B${\text{v}}_{\text{B x direction}}\text{ = }\frac{2-(0.5\times 1.73992)}{0.3}{\text{ = 3.7668 m s}}^{-1}$

${\text{v}}_{\text{B y direction}}\text{ = }\frac{0.9-(0.5\times 2.83929)}{0.3}{\text{ = -1.7322 m s}}^{-1}$

Step 6: Calculate resultant speed and direction of puck B after collision

$\text{speed = }\sqrt{{\text{v}}_{\text{x}}^2+{\text{v}}_{\text{y}}^2}=\sqrt{{3.7668}^2+{1.7322}^2}{\text{ = 4.15 m s}}^{-1}$

$\text{direction = }{\text{tan}}^{-1}\left(\frac{1.7322}{3.7668}\right)\text{ = 24.7°}$ (below the horizontal)