Momentum relative to the Speed of Light

The page focuses on momentum when traveling close to the speed of light

The Main Idea

Momentum is a property of a moving body; it can be narrowed down to simply mass and velocity (hence, [math]\displaystyle{ \overrightarrow{p} = m*\overrightarrow{v} }[/math]). However, the usual momentum equation does not always apply. When traveling near the speed of light, a new equation must be used. This equation was discovered by Albert Einstein in the early 1900's. This discovery revolutionized physics and introduced a new constant, gamma or [math]\displaystyle{ \gamma }[/math], a quantity relating velocity and momentum.

A Mathematical Model

The relative equation for momentum is as follows:
[math]\displaystyle{ \overrightarrow{p} = \gamma * m * \overrightarrow{v} }[/math]
where p is the momentum of the system, m is mass, and v is the velocity. The new constant [math]\displaystyle{ \gamma }[/math] is a bit more complicated.

The equation for [math]\displaystyle{ \gamma }[/math] is as follows:
[math]\displaystyle{ \gamma = \sqrt{\frac{1}{1-\frac{\left\vert \overrightarrow{v} \right\vert^2}{c^2}}} }[/math]
where v is again the velocity, and c is the speed of light or [math]\displaystyle{ 3 * 10^8 }[/math]

Again, this formula should only be used when traveling close to the speed of light. As you can see in the following chart, momentum is only noticeably affected around [math]\displaystyle{ 10^7 }[/math].

Lastly, momentum is most practical in the case of predicting position using Iterative Prediction.

Iterative prediction normally uses the position update equation: [math]\displaystyle{ {\vec{r}_{f} = \vec{r}_{i} + \vec{v}_{avg}{Δt}} }[/math]

This equation still applies relative to the speed of light, but appears in a slightly different form seen below.

[math]\displaystyle{ {\vec{r}_{f} = \vec{r}_{i} + \frac{1}{\sqrt{1-\frac{\vec{v}_{avg}^2}{c^2}}}\vec{v}_{avg}{Δt}} }[/math]

Examples

Simple

Suppose that a proton (mass = [math]\displaystyle{ 1.7 * 10^{-27} }[/math] kg) is moving with a velocity [math]\displaystyle{ .97c }[/math]

What is the momentum of the proton?

[math]\displaystyle{ \frac{v}{c} = \frac{.97c}{c} }[/math] [math]\displaystyle{ = .97 }[/math]

[math]\displaystyle{ \gamma = \frac{1}{\sqrt{1-(.97)^2}} = 4.1135 }[/math]

Plug values in

[math]\displaystyle{ p = \gamma * m*v = (4.1135)*(1.7*10^{-27})*(.97*(3*10^8)) = 2.035 * 10^{-18} }[/math] kgm/s

Middling

Suppose that a proton (mass = [math]\displaystyle{ 1.7 * 10^{-27} }[/math] kg) is moving with a velocity [math]\displaystyle{ \lt 1 * 10^7 , 2 * 10^7 , 3 * 10^7\gt }[/math] m/s.

What is the momentum of the proton?

[math]\displaystyle{ \left\vert \overrightarrow{v} \right\vert = \sqrt{(1*10^7)^2 + (2*10^7)^2 +(3*10^7)^2} }[/math] m/s [math]\displaystyle{ = 3.7 * 10^7 }[/math] m/s.

[math]\displaystyle{ \frac{\left\vert \overrightarrow{v} \right\vert}{c} = \frac{3.7 * 10^7 m/s}{3 * 10^8 m/s} }[/math] [math]\displaystyle{ = .12 }[/math]

[math]\displaystyle{ \gamma = \frac{1}{\sqrt{1-(.12)^2}} = 1.007 }[/math]

[math]\displaystyle{ \overrightarrow{p} = \gamma * m * \overrightarrow{v} }[/math]

Plug values in

[math]\displaystyle{ \overrightarrow{p} = (1.007)*(1.7*10^{-27})*\lt 1 * 10^7 , 2 * 10^7 , 3 * 10^7\gt = \lt 1.7 * 10^{-20}, 3.4 * 10^{-20}, 5.1 *10^{-20}\gt }[/math] kgm/s

Difficult

Connectedness

This concept of relativistic momentum affects several different majors. For example, on a quantum physics level, relativistic momentum aids in accurately predicting the position of a particle through a certain period of time.

Relativistic momentum is applicable throughout majors. Hopefully, one day, you'll find its use for you.

History

In 1905, Albert Einstein released his Special Theory of Relativity to the public. This theory set the "speed limit" for the universe at the speed of light. When objects, came closer to the speed of light, many entities are drastically changed (one being momentum). This changed the direction of physics immensely. After this discovery, physicists were better able to predict and calculate momentum on the microscopic scale of fast-moving particles.

Momentum relative to the Speed of Light

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