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Understanding Momentum
The Physics of Motion
Standard Formula
For most everyday objects, momentum ($p$) is simply the product of mass ($m$) and velocity ($v$).
Relativistic Formula
At high speeds (near light speed $c$), we must account for relativity using the Lorentz factor ($\gamma$).
Did you know?
Momentum is a vector quantity, meaning it has both magnitude and direction. In a closed system, total momentum is always conserved.
Relativistic Physics
As objects approach the speed of light (c), linear momentum $p=mv$ becomes inaccurate. We must use $p=\gamma mv$ where $\gamma$ is the Lorentz factor.
Kinetic Energy
While momentum is $m \times v$, kinetic energy is $0.5 \times m \times v^2$. Doubling velocity doubles momentum but quadruples the energy!
Key Concepts
Conservation of Momentum
One of the most important laws in physics. In a closed system where no external forces act, the total momentum before an event (like a collision or explosion) must equal the total momentum after. This is why a cannon recoils when fired—the forward momentum of the ball is balanced by the backward momentum of the cannon.
Impulse and Force
To change an object's momentum, you must apply a force over a period of time. This change in momentum is called Impulse ($J$). $$J = F \Delta t = \Delta p$$ This explains why airbags save lives: they increase the time ($\Delta t$) of the collision, which reduces the force ($F$) needed to stop your body's momentum.
Momentum vs. Kinetic Energy
While both depend on mass and velocity, they are different quantities. Momentum is a vector (has direction) and scales linearly with velocity. Kinetic Energy is a scalar (no direction) and scales with the square of velocity. This means a car moving at $2x$ speed has $2x$ momentum but $4x$ the destructive energy.