4.6 Collisions in Two Dimensions

Momentum is conserved in each direction

In 2D collisions, total momentum is conserved separately along the x-axis and y-axis. Break each object's velocity into components before and after the collision, then apply conservation of momentum independently to each axis: Σp_x,initial = Σp_x,final and Σp_y,initial = Σp_y,final.

Setting up the problem

Choose a coordinate system, often with the incoming object moving along the x-axis. Use sine and cosine to resolve velocities into components at given angles. After solving for final velocity components, recombine them using the Pythagorean theorem and inverse tangent to find the resulting speed and direction.

Elastic vs. inelastic in 2D

Just like 1D collisions, 2D collisions can be elastic (kinetic energy conserved) or inelastic (kinetic energy not conserved). Momentum conservation always holds regardless of collision type, but kinetic energy conservation only applies to elastic collisions — this can be used as an extra check or equation when needed.

Key terms

How is momentum conserved in 2D collisions?

Separately along each axis: total x-momentum before = after, and total y-momentum before = after.

How do you resolve a velocity into x and y components?

vx = v cos(θ), vy = v sin(θ), where θ is measured from the x-axis.

How do you find resultant speed from vx and vy?

v = √(vx² + vy²), using the Pythagorean theorem.

How do you find the direction of a resultant velocity?

θ = tan⁻¹(vy / vx), the inverse tangent of the y- and x-components.

What's the key difference between elastic and inelastic 2D collisions?

Elastic collisions conserve both momentum and kinetic energy; inelastic collisions conserve momentum only.

Try an AP-style question

In a 2-D collision, how is momentum conserved?

1. Only in the direction of the fastest object
2. Separately in the x and y directions
3. Only in the x direction
4. Only as total speed