# Conservation of Momentum in Two Dimension Scenarios

This topic is part of the HSC Physics course under the section Momentum, Energy and Simple Systems.

### HSC Physics Syllabus

• conduct an investigation to describe and analyse one-dimensional (collinear) and two-dimensional interactions of objects in simple closed systems (ACSPH064)
• analyse quantitatively and predict, using the law of conservation of momentum Σmv_{\text{before}}= Σmv_{\text{after}} and, where appropriate, conservation of kinetic energy Σ1/2mv_{\text{before}}= Σ 1/2 mv_{\text{after}}, the results of interactions in elastic collisions (ACSPH066)

### The Principle of Conservation of Momentum

The principle of conservation of momentum states that the total momentum of an isolated system remains constant if no external forces act on it. This principle applies in one, two, and three dimensions, and it's a direct consequence of Newton's third law of motion, which states that for every action, there's an equal and opposite reaction.

For two-dimensional motion, this conservation principle can be applied separately to each direction (usually defined as the x and y directions). The total momentum before the event (like a collision or an explosion) must equal the total momentum after the event, for each dimension.

$$\sum \vec{p}_{\text{initial}} = \sum \vec{p}_{\text{final}}$$

In the x-direction:

$$\sum m_i \times v_{i_{x_{\text{initial}}}} = \sum m_i \times v_{i_{x_{\text{final}}}}$$

In the y-direction:

$$\sum m_i \times v_{i_{y_{\text{initial}}}} = \sum m_i \times v_{i_{y_{\text{final}}}}$$

Solutions to the following examples can be found in the video above.

### Example 1

A 5 kg ball strikes a 10 kg stationary ball at 6 m s–1. After the collision, the 10 kg ball and 5 kg ball move at an angle of 30º and 60º above and below the x-axis as shown. What are the final speeds of the two balls?

### Example 2

A stationary bomb shell exploded to produce three fragments as shown. What is the velocity of fragment C?