The momentum p of an electron at velocity v near the...

The momentum $p$ of an electron at velocity $v$ near the velocity $c$ of light increases according to the formula $$ p=\frac{m_{0} v}{\sqrt{1-v^{2} / c^{2}}} $$ where $m_{0}$ is a constant (the rest mass). If an electron is subject to a constant force $F$, Newton's second law describing its motion is $$ \frac{d p}{d t}=\frac{d}{d t}\left(\frac{m_{0} v}{\sqrt{1-v^{2} / c^{2}}}\right)=F $$ Find the velocity as a function of time and show that the limiting velocity as $t$ tends to infinity is $c$. Find the distance traveled by the electron in time $t$ if it starts from rest.

The momentum $p$ of an electron at velocity $v$ near the velocity $c$ of light increases according to the formula
$$
p=\frac{m_{0} v}{\sqrt{1-v^{2} / c^{2}}}
$$
where $m_{0}$ is a constant (the rest mass). If an electron is subject to a constant force $F$, Newton's second law describing its motion is
$$
\frac{d p}{d t}=\frac{d}{d t}\left(\frac{m_{0} v}{\sqrt{1-v^{2} / c^{2}}}\right)=F
$$
Find the velocity as a function of time and show that the limiting velocity as $t$ tends to infinity is $c$. Find the distance traveled by the electron in time $t$ if it starts from rest.

Key Concepts

Hyperbolic Functions and Asymptotic Analysis

Hyperbolic functions naturally emerge in the solution of relativistic motion problems because they elegantly capture the asymptotic behavior mandated by special relativity. For instance, the velocity of an object under constant force can often be expressed in terms of the hyperbolic tangent function, which asymptotically approaches one. When multiplied by the speed of light, this result shows that the velocity levels off at the speed of light as time tends to infinity. Similarly, the integration involved in finding displacement results in expressions with hyperbolic functions, providing insight into the long-term behavior of relativistic systems.

Newton’s Second Law in Relativistic Dynamics

Newton’s second law, which equates force to the rate of change of momentum, retains its form in relativistic mechanics but requires using the relativistic expression for momentum. This leads to a differential equation whose solution describes how an object’s velocity evolves under a constant force, taking into account the relativistic increase in inertia. This formulation is key in analyzing how acceleration diminishes as an object’s speed nears the speed of light.

Differential Equations in Relativistic Motion

The dynamics of a particle under a constant force in a relativistic context involves solving a differential equation derived from the time rate of change of relativistic momentum. This process involves the application of calculus, particularly the techniques of differentiating composite functions and integrating functions that include the Lorentz factor. Solving these differential equations allows one to derive explicit expressions for velocity as a function of time and for the displacement over time.

Special Relativity

Special relativity redefines the relationship between space, time, mass, and energy at velocities approaching the speed of light. It introduces corrections to classical mechanics, such as time dilation and length contraction, and requires modifications to the definitions of momentum and energy. These corrections become significant as an object’s velocity gets closer to the speed of light, ensuring that no object can exceed this ultimate speed limit.

Relativistic Momentum

Relativistic momentum is a modification of the classical momentum formula to account for the effects described by special relativity. Instead of simply being the product of rest mass and velocity, momentum is scaled by the Lorentz factor, which increases sharply as the velocity approaches the speed of light. This adjustment guarantees that the momentum tends to infinity as the velocity tends to the speed of light, reflecting the impossibility of accelerating a massive object to or beyond that limit.

Transcript

00:02 All right.

00:04 So this is kind of an involved question because it has, you know, square roots and squares and whatnot.

00:12 But let's see.

00:13 So we are given the momentum of an electron.

00:16 Okay.

00:16 And near the speed of light, the momentum is something like this.

00:21 So if you remember, momentum is just given by m times the velocity, right? but what happens in special relativity, we get another factor, which is this square root factor.

00:32 If you can, you can't.

00:33 See in the denominator and this is the factor that we get near the velocity of light or the speed of light c well no need to go into that direction we're given the momentum p and what we're given is the newton second law which is given as d over d t of m not v over is equal to f okay so basically we're just given the uh the force as the time derivative of the momentum right and we've just used the form of the momentum and the equation asks us find the velocity as a function of time okay let's just stop right there velocity is a function of time and show that something something something something but let's try and find out the velocity of as a function of time how do we do that that's easy we just integrate both sides right so what we can do is we multiply both sides by d t and okay let me write this down multiply both sides by d t so we'll get a d not over m0 v over one minus v squared over c squared equals f d t okay and since f is constant a constant force if you can see here f is constant that means the right hand side is just it's a constant times d t and the left hand side is just d times something or it's not really d times but it's it's just a differential right so what we do is we integrate both sides that's it we just integrate both sides because both of them contain a differential right of some sort so when we do that the left side is just easy right i mean think about this dx integration is just x so we have a differential of x and if we integrated we get x if we indefinite integrated we get x similarly, if you have a differential of this thing, then the integration would just give us the same thing.

03:02 On the right -hand side, we'll have the integration, something like this...

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