state the principle of conservation of momentum

Linear Momentum and its Conservation

Learning Objective

State the principle of conservation of momentum.

1. Definition of Linear Momentum

Linear momentum, denoted by $\mathbf{p}$, is a vector quantity defined for a particle of mass $m$ moving with velocity $\mathbf{v}$:

$$\mathbf{p}=m\mathbf{v}$$

• Direction of $\mathbf{p}$ is the same as the direction of $\mathbf{v}$.
• SI unit: kilogram metre per second (kg·m·s\(^{-1}\)).

2. Impulse–Momentum Theorem

The change in momentum of an object is equal to the impulse applied to it.

$$\Delta\mathbf{p} = \mathbf{J} = \int \mathbf{F}\,dt$$

For a constant force $F$ acting over a time interval $\Delta t$:

$$\Delta\mathbf{p}=F\Delta t$$

3. Principle of Conservation of Momentum

Statement: In an isolated system (i.e., a system on which no external net force acts), the total linear momentum remains constant.

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

This principle follows directly from Newton’s second and third laws and is valid for any interaction, whether the bodies collide or separate.

4. Conditions for Conservation

1. The system must be closed – no mass enters or leaves.
2. The net external force on the system must be zero (or negligible during the interaction).
3. All internal forces are equal in magnitude and opposite in direction (Newton’s third law).

5. Types of Collisions

Collision Type	Momentum	Kinetic Energy	Typical Example
Elastic	Conserved	Conserved	Billiard balls striking each other
Inelastic	Conserved	Not conserved (some lost as deformation, heat, sound)	Car crash where vehicles stick together
Perfectly Inelastic	Conserved	Maximum loss of kinetic energy	Two clay balls sticking together after impact

6. Example Problem

Problem: A 2.0 kg cart moving at $3.0\ \text{m s}^{-1}$ collides head‑on with a 3.0 kg cart initially at rest. After the collision the two carts stick together. Find their common speed immediately after the collision.

Solution:

1. Write the conservation of momentum equation: $$m_1 v_{1i} + m_2 v_{2i} = (m_1+m_2)v_f$$
2. Substitute the known values: $$ (2.0)(3.0) + (3.0)(0) = (2.0+3.0)v_f $$
3. Solve for $v_f$: $$6.0 = 5.0 v_f \quad\Rightarrow\quad v_f = 1.2\ \text{m s}^{-1}$$

The combined system moves at $1.2\ \text{m s}^{-1}$ in the original direction of the 2.0 kg cart.

7. Suggested Diagram

8. Summary

• Linear momentum $\mathbf{p}=m\mathbf{v}$ is a conserved vector quantity in isolated systems.
• Conservation of momentum is a powerful tool for analysing collisions and explosions.
• Only external forces can change the total momentum of a system; internal forces cancel out.