Conservation of momentum in two dimensions.

GET THE FACTS:

• Momentum is conserved in all collisions in one dimension.

• Momentum is a vector quantity and the above principle applies when the objects collide at other angles to each other.

• The resultant momentum of the colliding objects is considered.

The next few slides show three methods of determining the

resultant momentum of objects colliding and moving in two

dimensions.

BY THE END OF THIS SECTION YOU SHOULD BE ABLE TO:

•Apply the Law of Conservation of Momentum to collisions in two dimensions.

•Determine the resultant of the vectors in these collisions using

• graphical methods

• the method of components

• trigonometric methods.

DETERMINING RESULTANT MOMENTUM:

GRAPHICAL METHOD

pball A + pball B = presultant(total momentum)

Select a suitable scale

Draw line A to represent the momentum of ball A

From the head of line A, draw a line to represent the momentum of ball B

Complete the triangle, and the third side represents the resultant (or total) momentum.

DETERMINING RESULTANT MOMENTUM

TRIGINOMETRIC METHOD

This is a right angle triangle.

Pball A = 10 cos 30o

P ball B = 10 sin 30o

Pball B = 10 sin 30o = 10 x 0,5 = 5,0 kg.m.s-1

Velocity of ball A = 2,5 m.s-1

Pball A = 10 cos 30o = 10 x 0,866 = 8,66 kg.m.s-1

Velocity of ball A = 4,33 m.s-1

DETERMINING RESULTANT MOMENTUM

COMPONENTS METHOD

Before the collision:

Components of p in y direction = 0

Components of p in x direction = px A + px B = (2 x 5) + 0

= 10 kg.m.s-1

ptotal = 10 kg.m.s-1 in x direction

After the collision:

Components of p in x direction:

px = px of A + px of B

= (2 x vA cos 30o) + (2 x vB cos 60o)

= 1,73vA + vB = 10 kg.m.s-1 ……….eqn 1

Components of p in the y direction:

py = py of A + py of B

= (2 x vA sin 30o) - (2 x vB sin 60o)

= vA - 1,73vB = 0 kg.m.s-1

vA = 1,73vB …………………………… eqn 2

Substitute in eqn 1:

(1,73 x 1,73vB) + vB = 10

vB = 2,5 m.s-1

Substitute in eqn 2:

vA = 1,73 x 2,5 = 4,33 m.s-1

Relative velocity in two dimensions A boat sails on a bearing of 0o at a speed of 25

m.s-1. An ocean current flows at 10m.s-1 on a bearing of 90o. Determine the resultant velocity of the boat.

Vresultant of the boatV of the boat

relative to still

water

V of the ocean current

Use graphical, trigonometric or component methods to determine the resultant velocity of the boat.