Revise Momentum for Cambridge 9231 (3.6)

Linear momentum is a fundamental concept, often described as the 'quantity of motion' an object possesses. It is a vector quantity, meaning it has both magnitude and direction. The direction of the momentum is the same as the direction of the object's velocity.

Momentum ($\vec{p}$) = Mass ($m$) $\times$ Velocity ($\vec{v}$)

The cornerstone of this topic is the Principle of Conservation of Linear Momentum. It states that for any system of colliding objects, the total momentum before the collision is equal to the total momentum after the collision, provided that no external forces (like friction or air resistance) act on the system. This is a direct consequence of Newton's Third Law.

For two particles in a direct collision: $m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$

When a force acts on an object, it changes its momentum. The impulse of a force is defined as the product of the force and the time interval over which it acts. Crucially, impulse is equal to the change in momentum it produces. This is particularly useful for situations involving large forces acting over very short times, such as a cricket bat hitting a ball, where the exact force profile is unknown.

Impulse ($\vec{I}$) = Change in Momentum ($\Delta \vec{p}$) \ $\vec{I} = m\vec{v} - m\vec{u} = \int F(t) \text{\textdollar}, dt$

For a constant force $F$ acting over a time $\Delta t$, this simplifies to $I = F \Delta t$. Graphically, the impulse is the area under a force-time graph. Since impulse is a vector, a positive impulse indicates a force acting in the positive direction, and a negative impulse indicates a force in the negative direction.

While momentum is conserved in a closed system, kinetic energy is usually not. Most real-world collisions are 'inelastic', meaning some kinetic energy is converted into heat, sound, or work done in deforming the objects. The coefficient of restitution, $e$, is a number that tells us how 'bouncy' a collision is. It's defined as the ratio of the relative speed of separation to the relative speed of approach.

$e = \frac{\text{Speed of Separation}}{\text{Speed of Approach}} = \frac{|v_2 - v_1|}{|u_1 - u_2|}$