An electron moves to the right with a speed of 0.90 c relative to the laboratory frame. A proton

step 1 Hello everyone, welcome to special activity. And another wonderful problem here. And this one, w

An electron moves to the right with a speed of 0.90 c...

Key Concepts

Special Relativistic Velocity Addition

This concept explains how velocities are combined when objects are moving at speeds approaching the speed of light. Unlike classical addition, the relativistic formula accounts for the invariant speed of light, ensuring that the resulting speed does not exceed this universal limit. It is derived from the Lorentz transformations and is essential for accurately describing high-speed motion.

Reference Frames

A reference frame is a coordinate system used by an observer to measure positions, velocities, and time intervals. In physics, especially in relativity, different observers may be in different inertial frames, which can lead to different measurements. Understanding how to transition between these frames is key to analyzing the motion of objects at high speeds.

Lorentz Transformation

The Lorentz transformation comprises a set of equations that relate the space and time coordinates of events as observed from different inertial frames moving relative to each other. It forms the mathematical foundation of special relativity, facilitating the derivation of effects such as time dilation, length contraction, and the relativistic addition of velocities.

Transcript

00:01 Hello everyone, welcome to special activity.

00:06 And another wonderful problem here.

00:10 And this one, we have two particles moving with respect to each other.

00:18 Let me draw them here.

00:21 The first one is the electron.

00:23 And we are given the velocity of this electron to be 0 .9c.

00:31 We have also a proton moving in the same direction this proton has a velocity of 0 .7c and in this problem we are required to get the velocity to the proton okay with respect to the left frame actually 0 .7c here is with respect to this electron so subpochre to write it as v electron just to make the notation here consistent this is the electron as respect to the lab frame this is the v proton with respect to the electron okay and the way i think about these problems is to write the the formula for the relative velocity the classical relative velocity so b protons with respect to the electron is what the velocity sorry of the product of the product this is classically defined as b a v proton with respect to the lap frame right remember this formula minus v pro the electron with respect to the lab frame and you can solve for b if you if you solve for b the the proton as respect to the left frame this one this will be what the proton with respect to the electron plus as you can see here they are in the same direction so no worries about the signs of the proton or electron that does take them to be positive here so the positive direction is this way and you have the proton with respect to the electron plus to the left frame.

02:32 And then comes the relativity part.

02:34 The relativity corrects this expression here by this factor.

02:41 So this is classically so far, this is classical analysis, and the relativity comes out to correct for this.

02:50 Let me call the b times ve, where vba with respect to the left frame and the...

03:00 With respect to the electron.

03:05 So actually let's write it down.

03:08 However, at the end, this times e, ve, with respect to l, divided by c square.

03:17 Wonderful...