Momentum And Collisions

A particle of mass m is moving at velocity v.

The linear momentum is defined as:

Impulse is defined as an average force F acting for a time Δt (this time is typically short). Mathematically, impulse is FΔt.

If an impulse acts on a particle of mass m, its mοmentum will change by an amount ΔP. We can express this as:

where Δv is the change in velocity of the particle.

Conservation Of Linear Mοmentum

When two objects collide, mοmentum is conserved. The figure below shows a collision between two objects (before and after).

Where:

V_a1 is the initial velocity of object a, before collision

m_a is the mass of object a

V_b1 is the initial velocity of object b, before collision

m_b is the mass of object b

V_a2 is the final velocity of object a, after collision

V_b2 is the final velocity of object b, after collision

The vector equation for conservation of linear mοmentum can be expressed as:

For all three directions in x, y, z this becomes:

For an elastic collision, kinetic energy is conserved. Thus,

For elastic collisions in one-dimension (head-on collision):

Conservation Of Angular Mοmentum

The angular mοmentum L for a body rotating about a fixed axis is defined as:

Where:

I is the rotational inertia of the body about the axis of rotation

w is the angular velocity of the body

If no net external torque acts on the body, L = constant.

Thus,

where the subscripts 1 and 2 denote "initial" and "final".

For example, a figure skater pulling his arms in during rotation will spin faster as a result. This is because there is no net external torque acting on his body. So as he pulls in his arms he is decreasing his I value, which results in w increasing (since his angular mοmentum L is constant).

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