Colliding Objects (1-D)


When objects collide, their momentum changes. Colliding objects are goverened by the law of conservation of momentum, stated as follows:

Law of Conservation of Momentum:

Provided no external forces act on the system, the total momentum of the system before any collision occurs is equal to the total momentum of the system after the collision

We can apply this law of collisions to determine the formula:

Total momentum before collision = Total momentum after collision

Or stated mathematically:

{ m }_{ 1 }{ u }_{ 1 }+{ m }_{ 2 }{ u }_{ 2 }={ m }_{ 1 }{ v }_{ 1 }+{ m }_{ 2 }{ v }_{ 2 }

Where

{ 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 }

Examples of collisions that we will consider:

  • A moving object collides with a stationary object and then stops
  • Objects moving in the same direction, colliding and sticking together
  • Objects moving in the same direction, colliding and moving in the same direction or rebounding
  • Objects moving in the opposite direction, colliding and moving in the same direction or rebounding

As momentum is a vector, in the equation above, initial and final velocities must take into account direction.


Example 1: A moving object collides with a stationary object and then stops

A car with a mass of 500kg travelling at 3\cfrac { m }{ s } to the north, collides with a 750kg which is stationary. After the collision the 500kg car stops. What happens to the 750kg car?

{ m }_{ 1 }500kg

{ u }_{ 1 }3\cfrac { m }{ s } to the north (note: north as positive)

{ v }_{ 1 } = 0\cfrac { m }{ s }

{ m }_{ 2 }750kg

{ u }_{ 2 }0\cfrac { m }{ s }

{ v }_{ 2 } = ?

Substituting values into:

{ m }_{ 1 }{ u }_{ 1 }+{ m }_{ 2 }{ u }_{ 2 }={ m }_{ 1 }{ v }_{ 1 }+{ m }_{ 2 }{ v }_{ 2 }

(500\times 3)+0=0+(750\times { v }_{ 2 })

1500=750{ v }_{ 2 }

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


Example 2: Objects moving in the same direction, colliding and sticking together

A shopping cart with a mass of 40kg travelling at 1.5\cfrac { m }{ s } east, collides with a 10kg shopping cart travelling at 2.5\cfrac { m }{ s } in the same direction. After the collision the trolleys were stuck together and moving at the same speed. What is the velocity of the combined masses after the collision.

{ m }_{ 1 }40kg

{ u }_{ 1 }1.5\cfrac { m }{ s } to the east (note: east as positive)

{ v }_{ 1 } = ?

{ m }_{ 2 }10kg

{ u }_{ 2 }2.5\cfrac { m }{ s }

{ v }_{ 2 } = ?

In problems where the objects join together after the collision, it is convenient to use the combined mass. We will write this as { m }_{ 3 }:

{ m }_{ 3 }40+10=50kg

{ v }_{ 3 } = ?

Substituting values into:

{ m }_{ 1 }{ u }_{ 1 }+{ m }_{ 2 }{ u }_{ 2 }={ m }_{ 3 }{ v }_{ 3 }

(40\times 1.5)+(10\times 2.5)=(50\times { v }_{ 3 })

85=50{ v }_{ 3 }

{ v }_{ 3 }=1.7\cfrac { m }{ s } east


Example 3: Objects moving in the same direction, colliding and moving in the same direction or rebounding

A 250g ice puck travelling at 5\cfrac { m }{ s } east, collides with an identical ice puck travelling in the same direction at 2.5\cfrac { m }{ s } . After the collision the first ice puck rebounds in the opposite direction at 1.5\cfrac { m }{ s } . What is the velocity of the second ice puck after the collision?

{ m }_{ 1 }250g

{ u }_{ 1 }5\cfrac { m }{ s } to the east (note: east as positive)

{ v }_{ 1 } = 1.5\cfrac { m }{ s } west = -1.5\cfrac { m }{ s }

{ m }_{ 2 }250g

{ u }_{ 2 }2.5\cfrac { m }{ s } to the east

{ v }_{ 2 } = ?

Substituting values into:

{ m }_{ 1 }{ u }_{ 1 }+{ m }_{ 2 }{ u }_{ 2 }={ m }_{ 1 }{ v }_{ 1 }+{ m }_{ 2 }{ v }_{ 2 }

(0.25\times 5)+(0.25\times 2.5)=(0.25\times -1.5)+(0.25\times { v }_{ 2 })

1.25+0.625=-0.375+0.25{ v }_{ 2 }

1.875=-0.375+0.25{ v }_{ 2 }

2.25=0.25{ v }_{ 2 }

{ v }_{ 2 }=9\cfrac { m }{ s } west


Example 4: Objects moving in the opposite direction, colliding and moving in the same direction or rebounding

A ball with a mass of 2kg travelling at 3\cfrac { m }{ s } to the north, collides with another ball with a mass of 4kg moving in the opposite direction at 3\cfrac { m }{ s } . After the collision the 2kg ball bounces in the opposite direction at 4\cfrac { m }{ s } . What happens to the 4kg ball?

{ m }_{ 1 }2kg

{ u }_{ 1 }3\cfrac { m }{ s } to the north (note: north as positive)

{ v }_{ 1 } = 4\cfrac { m }{ s } to the south = -4\cfrac { m }{ s }

{ m }_{ 2 }4kg

{ u }_{ 2 } =3\cfrac { m }{ s } to the south = -3\cfrac { m }{ s }

{ v }_{ 2 } = ?

Substituting values into:

{ m }_{ 1 }{ u }_{ 1 }+{ m }_{ 2 }{ u }_{ 2 }={ m }_{ 1 }{ v }_{ 1 }+{ m }_{ 2 }{ v }_{ 2 }

(2\times 3)+(4\times -3)=(2\times -4)+(4\times { v }_{ 2 })

6-12=-8+4{ v }_{ 2 }

2=4{ v }_{ 2 } 

{ v }_{ 2 }=0.5\cfrac { m }{ s } north