Space pilot Mavis zips past Stanley at a constant speed...

Space pilot Mavis zips past Stanley at a constant speed relative to him of 0.800c. Mavis and Stanley start timers at zero when the front of Mavis's ship is directly above Stanley. When Mavis reads 5.00 s on her timer, she turns on a bright light under the front of her spaceship. (a) Use the Lorentz coordinate transformation derived in Example 37.6 to calculate x and t as measured by Stanley for the event of turning on the light. (b) Use the time dilation formula, Eq. (37.6), to calculate the time interval between the two events (the front of the spaceship passing overhead and turning on the light) as measured by Stanley. Compare to the value of $t$ you calculated in part (a). (c) Multiply the time interval by Mavis's speed, both as measured by Stanley, to calculate the distance she has traveled as measured by him when the light turns on. Compare to the value of $x$ you calculated in part (a).

Space pilot Mavis zips past Stanley at a constant speed relative to him of 0.800c. Mavis and Stanley start timers at zero when the front of Mavis's ship is directly above Stanley. When
Mavis reads 5.00 s on her timer, she turns on a bright light under the front of her spaceship. (a) Use the Lorentz coordinate transformation derived in Example 37.6 to calculate x and t as measured by Stanley for the event of turning on the light. (b) Use the time dilation formula, Eq. (37.6), to calculate the time interval between the two events (the front of the spaceship passing overhead and turning on the light) as measured by Stanley. Compare to the value of $t$ you calculated in part (a). (c) Multiply the time interval by Mavis's speed, both as measured by Stanley, to calculate the distance she has traveled as measured by him when the light turns on. Compare to the value of $x$ you calculated in part (a).

Key Concepts

Relativistic Kinematics

Relativistic kinematics deals with the description of motion when velocities are close to the speed of light. This includes applying concepts like Lorentz transformation and time dilation to calculate distances, time intervals, and positions of events, ensuring that the results are consistent with the invariance of the speed of light across all inertial frames.

Inertial Reference Frames

Inertial reference frames are coordinate systems in which an object either remains at rest or moves at a constant velocity unless acted upon by a force. The principle of relativity holds in these frames, meaning that the laws of physics, including the formulation of the Lorentz transformation and the phenomenon of time dilation, are the same in all inertial frames.

Time Dilation

Time dilation is a relativistic phenomenon where a clock moving relative to an observer ticks slower compared to a clock at rest with respect to that observer. It plays a crucial role in analyzing scenarios in special relativity, where the proper time measured in one frame differs from the time observed in another moving frame.

Lorentz Transformation

The Lorentz transformation provides the mathematical framework used to relate space and time coordinates of events as measured in different inertial frames that are moving at a constant velocity relative to one another. It accounts for effects like time dilation and length contraction and is fundamental in solving problems in special relativity involving events occurring in different reference frames.

Transcript

00:01 This is chapter 37 problem number 16.

00:03 We have two ships operated by mavis and stanley.

00:09 So mavis is passing stanley by the speed of 0 .8c.

00:15 And when that's happening, when she appears to be overhead, let's say this is mavis is free.

00:20 This is stanley's.

00:22 Mavis is moving to 0 .8 at a speed of 0 .8c.

00:27 So for mavis, x prime is 0 .0.

00:31 And t prime, the time that you measures is five seconds.

00:36 And through lawrence transformation in part a, we're asked to calculate what the distance would look like based on stanley's reference frame.

00:47 So for that, we're going to use our lawrence transformation for both position and time from the example on your book 37 .7.

00:57 If you can take a look at that part.

01:00 X prime the position as measured in mavisus frame plus u times t prime this gives the position for stately and the time would be gamma factor t prime plus u x prime over c squared now all we need to do is plug in the numbers into the calculation um let's calculate what gamma factor is because we're going to be using it over over again one minus 0 .8c squared over c squared.

01:32 What we get heard here is 1 .6666.

01:36 A lot of six is seven.

01:37 So let's write it in terms of 1 .6 .7.

01:40 Okay.

01:41 Let's roll it up...

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