Colliding Objects (2-D)


Collisions often occur between objects that result in the objects moving off in different directions. Similarly to collision in 1-D:

{ m }_{ 1 } = mass of object 1 in kg

{ u }_{ 1 } = initial velocity object 1 in \cfrac { m }{ s }

{ v }_{ 1 } = final velocity object 1 in \cfrac { m }{ s }

{ m }_{ 2 } = mass of object 2 in kg

{ u }_{ 2 } = initial velocity object 2 in \cfrac { m }{ s }

{ v }_{ 2 } = final velocity object 2 in \cfrac { m }{ s }

Consider the diagram below:

An object moving to the right collides with another stationary object and they move off in different directions to the original direction. The new directions are noted as { \theta }_{ 1 } and { \theta }_{ 2 } and are in relation to the original direction.

Assuming no external forces act on the system, momentum will be conserved. These problems are resolved by considering the horizontal and vertical components of momentum and applying the law of conservation of momentum.

x- components: { m }_{ 1 }{ u }_{ 1 }={ m }_{ 1 }{ v }_{ 1 }cos{ \theta }_{ 1 }+{ m }_{ 2 }{ v }_{ 2 }cos{ \theta }_{ 2 } 

y- components: 0={ m }_{ 1 }{ v }_{ 1 }sin{ \theta }_{ 1 }+{ m }_{ 2 }{ v }_{ 2 }sin{ \theta }_{ 2 } 


Example 1:

An 400g pool ball moving to the right at 2.5\cfrac { m }{ s } collides with an identical stationary pool ball and they move off in different directions to the original direction. The original pool ball moves off at 1.6\cfrac { m }{ s } . The new directions are noted as 30^{ \circ } and 35.54^{ \circ } as shown below:

What is the velocity of the second pool ball?

Either method below will determine the speed of the second pool ball. However, by calculating both x and y components we can also demonstrate that momentum was conserved in this collision. 

x-components: 

{ m }_{ 1 }{ u }_{ 1 }={ m }_{ 1 }{ v }_{ 1 }cos{ \theta }_{ 1 }+{ m }_{ 2 }{ v }_{ 2 }cos{ \theta }_{ 2 } 

0.4\times 2.5=0.4\times 1.6cos30^{ \circ }+0.4\times { v }_{ 2 }cos35.54^{ \circ }

1=0.55+0.33{ v }_{ 2 }

0.45=0.33{ v }_{ 2 }

{ v }_{ 2 }=1.38\cfrac { m }{ s }

y-components:

0={ m }_{ 1 }{ v }_{ 1 }sin{ \theta }_{ 1 }+{ m }_{ 2 }{ v }_{ 2 }sin{ \theta }_{ 2 }

0=0.4\times 1.6sin30^{ \circ }+0.4\times { v }_{ 2 }sin35.54^{ \circ }

0=0.32+0.23{ v }_{ 2 }

-0.32=0.23{ v }_{ 2 }

{ v }_{ 2 }=1.38\cfrac { m }{ s }