Linear Momentum

(Linear) Momentum is a measurement of mass in motion: how much mass is in how much motion. It is usually given the symbol p.

By definition, p = m * v

Where m is the mass and v is the velocity. The standard units for momentum are kg⋅m/s, and momentum is always a vector quantity.

Impulse

Impulse is the integral of a force, F, over the time interval, t, for which it acts. Impulse is basically change in momentum.

For a constant force, impulse J = F * Δt  = Δp

This is because Δp = m * Δv = m * a * Δt = F * Δt

From above, we should also note that F = Δp / Δt. In other words, force equals rate of linear momentum change.

Elastic vs Inelastic Collision

A perfectly elastic collision is defined as one in which there is no loss of kinetic energy in the collision. An inelastic collision is one in which part of the kinetic energy is changed to some other form of energy in the collision.

Law of conservation of linear momentum

Wether the collision is elastic or inelastic, in an isolated system, the total momentum before the collision is equal to the total momentum after the collision.

m₁v_1i + m₂v_2i = m₁v_1f + m₂v_2f

You can derive the law of conservation of linear momentum from Newton's 3rd law of action-reaction:

Let F₁₂ be the force that m₁ exerts on m₂, and let F₂₁ be the force that m₂ exerts on m₁. According to Newton's 3rd law of action-reaction, we have

F₁₂ = - F₂₁

F₁₂ + F₂₁ =0

Since F₁₂ = Δp₁ / Δt, and F₂₁ = Δp₂ / Δt, we have

Δp₁ / Δt + Δp₂ / Δt = 0

Δ(p₁+p₂) / Δt = 0

The change in the sum of the total momentum for the system, Δ(p₁+p₂), is therefore 0, and the sum of the total momentum for the system, p₁+p₂, is therefore a constant

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