Elastic Collisions in Two Dimensions: Oblique Impacts

Elastic Collisions in Two Dimensions: Oblique Impacts

Conservation of Linear Momentum

• Linear momentum is conserved in collisions, regardless of whether it’s elastic or inelastic.
• The total momentum before a collision is equal to the total momentum after the collision in the absence of external forces.
• In two dimensions, momentum is conserved independently in each direction.

Coefficient of Restitution

• The coefficient of restitution (e) is a measure of elasticity of a collision.
• For perfectly elastic collisions, e = 1 and for perfectly inelastic collisions, e = 0.

Specifics of Two-Dimensional Collisions

• When two bodies collide obliquely, the collision can be split into two components, the line of centers and perpendicular to line of centers.
• After the collision, they move off in separate directions, making different angles with the original line of motion.
• These angles can be determined using their initial velocities and the principle of conservation of momentum.

Calculating Final Velocities

• The angles and velocities of the bodies after the collision can be calculated using conservation of momentum and the coefficient of restitution.
• The procedure involves splitting the velocities into components, considering momentum conservation separately for each direction, and using the restitution equation in the direction along the line of centers.

Special Cases of Oblique Impact

• When two bodies of equal mass collide and one is initially at rest, they will move off at right angles after a perfectly elastic collision.
• When a body collides obliquely with a wall, it will bounce off with the same speed and an angle of reflection equal to the angle of incidence in a perfectly elastic collision.