Inelastic Collisions in Two Dimensions

Two cars approach an intersection at a 90o angle and collide inelastically, sticking together after the collision. What is the velocity (speed and direction) of the two-car clump of twisted metal immediately after the collision?

Sounds like this could develop into a complicated problem, doesn't it? Well, there's good news. It isn't a complicated problem, because the velocity of the cars after the collision has to be the same as the velocity of the center of mass of the two-car system immediately before the collision. In other words, a two-dimensional inelastic collision solves exactly like a one-dimensional inelastic collision, except for one additional easy calculation. Viewed from the center of mass, all inelastic collisions look alike!

Example 4: A Collision at an Intersection

A 1000 kg car is moving eastward at 20 m/s. It collides inelastically with a 1500 kg van traveling northward at 30 m/s. What is the velocity of the two vehicles immediately after the collision?


problem diagramIt is an easy, straightforward problem to find the velocity of the center of mass of the two-car system immediately after the collision. First, the x- and y-components of the velocity of the center of mass:

vx = 8 m/s, vy = 18 m/s

knowing the x- (eastward) and y- (northward) components of the velocity of the center of mass, the magnitude is:

v_sub_cm = 19.7 m/s

Also, the angle that the velocity vector makes with the x- (eastward) axis is a routine calculation:

theta = 66 degrees

Since the velocity of the system's center of mass is unaltered by the collision, the two vehicles have a velocity of 19.7 m/s at an angle of 66o north of east immediately after the collision.

last update March 13, 2006 by JL Stanbrough