In a high-speed football game, a running back traveling at 55.0 % the speed of light relative to

In a high-speed football game, a running back traveling at...

Key Concepts

Relativistic Velocity Addition

This concept refers to the method used in special relativity to combine velocities that are significant fractions of the speed of light. Unlike classical addition of speeds, relativistic velocity addition accounts for time dilation and length contraction effects, ensuring that no relative velocity exceeds the speed of light. It provides a formula that correctly transforms speeds observed in one inertial frame to another.

Invariance of the Speed of Light

One of the foundational principles of special relativity is that the speed of light in a vacuum is constant and is perceived to be the same (approximately 3.00 x 10^8 m/s) regardless of the relative motion of the source and the observer. This principle ensures that light always travels at the same speed in all inertial reference frames, which has profound implications for how events are observed in different relativistic contexts.

Reference Frames in Special Relativity

Reference frames are coordinate systems used to describe the positions and motions of objects in space and time. In relativistic scenarios, different observers moving at high speeds relative to each other will measure time and space differently. The transformation between these frames, governed by the Lorentz transformation, is essential for understanding how various phenomena such as time dilation, length contraction, and velocity addition occur.

Transcript

00:01 In this problem, you have a running back, you have a receiver, and then you have a football that's being thrown between the two.

00:11 Now, initially, you're given the information relative to the field.

00:15 This is in the perspective of the field.

00:18 You have the receiver running, 0 .65c relative to the field, the running back, 0 .55c.

00:24 And now in the end, we want the, i don't include the ball here as yet, we want the balls, velocity relative to the receiver.

00:34 That's our end.

00:36 But there's a couple of steps they have to be gone through to get to that point.

00:40 Now, the first thing we have to do is let's find the velocity of the running back relative to the receiver.

00:52 Now, let me write this out and i'll have a few things to say.

00:56 Running back prime.

00:57 That means relative to the s prime frame, which is the receiver.

01:01 Is going to be u the running back minus v, 1 minus urb, v over c square.

01:16 Now, one thing i want to say, you'll see me do it in the next part of this problem, the last part of this part a, really.

01:23 When do you use the minus form of the u, the addition of velocity formula, and when do you use the plus form? the minus form is used when in your picture, what you're looking for is the velocity relative to the frame that is moving.

01:43 The receiver, the s prime frame is moving, so we use the one with the minus signs.

01:49 In my next picture, a picture down here, you'll see i'm going to be doing it in terms of the frame that is not moving, so i'll be using it with the plus signs.

01:59 So minus sign goes with the moving frame in the picture.

02:04 And the plus sign goes with the frame that is not moving.

02:08 The perspective of the picture is for it.

02:10 See, the perspective of this picture is relative to the field.

02:15 So we can put in these values, and this will allow us to draw another picture and remove the field completely from it.

02:22 So 0 .55c, 0 .65c, 1 minus 0 .55c, 0 .65c, 0 .6c, 0 .6c.

02:37 Divided by c squared, so the c squares go away in the bottom.

02:42 And this works out to be minus 0 .156c, which makes sense.

02:50 Look at what's going on here relative to the field.

02:53 The receiver is moving 0 .65 .55.

02:58 So you expect this to be negative.

03:14 Let's now do this with now the running back being, a frame.

03:23 So this is a running back frame now.

03:26 I'll call this s double prime.

03:28 We have a ball.

03:42 Now, this is going to be from the perspective of the receiver from the s prime frame.

03:48 So i got an s, and s prime, and s double prime frame.

03:52 So when i do that in this picture technically, the velocity i'm drawing is relative to the frame without the arrow that's at rest.

04:01 This frame then is the running back frame relative to s prime, which you just found.

04:10 So v running back, i'll qualify it because there's so many different symbols here, 1 .0 .156c.

04:21 Negative.

04:23 Remember, in generally, these are not speeds.

04:27 They really are one -dimensional velocities.

04:31 Now, i'm just going to put in here just to remind us that we're told the ball relative to the running back is 0 .8c.

04:41 I put it in parentheses because it really does not belong here.

04:44 This is intended to be for as relative to s prime...

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