Two pucks collide on smooth ice and skid away at angles. Nothing about the crash looks one-dimensional anymore. Does the conservation law survive the second dimension?
It does, and in a stronger form than you might expect: the bookkeeping runs once for each direction. Pick two perpendicular directions, east-west and north-south, and track them separately.
Here is the scene on smooth ice, taking east and north as the positive directions: a 1 kg puck slides east at 4 m/s, and a second 1 kg puck slides north at 3 m/s. They collide off-center and both change direction.
Before the collision, the system's eastward momentum totals 4 kg·m/s, all of it the first puck's. The northward momentum totals 3 kg·m/s, all of it the second puck's.
After the collision, the sensors read: the first puck carries 2.5 kg·m/s east and 1.2 kg·m/s north; the second carries 1.5 kg·m/s east and 1.8 kg·m/s north.
Total the east column: 2.5 + 1.5 = 4 kg·m/s. Total the north column: 1.2 + 1.8 = 3 kg·m/s. Each direction's total is exactly what it was.
The two directions never trade: eastward momentum cannot become northward momentum, however the pucks spin away. Each direction keeps its own books.
So a two-dimensional problem is two one-dimensional problems sharing a page: one conservation statement for east-west, one for north-south.
Take east and north as the positive directions. Before a collision on smooth ice, a system's total eastward momentum is 9 kg·m/s. The two objects collide and both change direction. What is the system's total eastward momentum after the collision?
No external force acts along the ice, so momentum is conserved along each direction separately.
The eastward total therefore keeps its value through the collision.
The system's total eastward momentum after the collision is 9 kg·m/s.
Take east and north as the positive directions. Before a collision, a system's total northward momentum is 5 kg·m/s. Afterward, one object carries 2 kg·m/s northward. How much northward momentum does the other object carry?
The northward total is conserved: it is still 5 kg·m/s after the collision.
One object carries 2 kg·m/s of it.
So the other object carries the rest: 5 − 2 = 3 kg·m/s.
The other object carries 3 kg·m/s northward.
Take east and north as the positive directions. Before a collision on smooth ice, a system's total eastward momentum is 10 kg·m/s and its total northward momentum is 4 kg·m/s. After the collision, one object carries 6 kg·m/s eastward. Enter the system's TOTAL eastward momentum after the collision in kg·m/s.
Two objects collide on smooth ice, moving in two dimensions. Which totals stay constant through the collision?
Piet says: 'In a two-dimensional collision on smooth ice, both pucks change direction, but the system's total northward momentum after equals the total northward momentum before.' Is he correct?
Take east and north as the positive directions. Before a collision on smooth ice, a system's total northward momentum is 6 kg·m/s. After the collision, puck A carries 2.5 kg·m/s northward. Enter the northward momentum puck B carries, in kg·m/s.
Which statement about a two-dimensional collision with no external force is correct?
Take east and north as the positive directions. Before a collision on smooth ice, cart A carries 7.2 kg·m/s eastward and cart B carries 1.8 kg·m/s eastward. After the collision, cart A carries 5.4 kg·m/s eastward. Enter cart B's eastward momentum after the collision, in kg·m/s.