James, Daniella, and Tara all possess identical clocks. As...

Key Concepts

Time Dilation

Time dilation is a fundamental aspect of special relativity which states that the time interval between two events as measured by an observer in motion relative to the events is longer than the interval measured by an observer at rest with respect to the events. This effect causes moving clocks to appear to run slower than those in the observer's own frame.

Relativistic Velocity Addition

Relativistic velocity addition describes how to correctly calculate the resultant velocity of one object as seen from the frame of another when both are moving with respect to a common inertial frame. Unlike classical addition of velocities, this formula accounts for the finite speed of light, ensuring that the combined speed never exceeds the speed of light.

Lorentz Factor

The Lorentz factor, denoted by ?, quantifies the degree of time dilation and length contraction experienced by objects moving at relativistic speeds. It is a critical factor in the Lorentz transformations which relate space and time coordinates between different inertial frames.

Inertial Reference Frames

Inertial reference frames are non-accelerating frames of reference in which the laws of physics, including the principles of special relativity, hold true. Observers in different inertial frames may measure different times and distances, and transformations such as time dilation and relativistic velocity addition are needed to relate these measurements.

Transcript

00:01 Hello, everyone.

00:01 In this problem, we're asked to calculate something pretty complicated.

00:06 So we're asked to find, given that we have three frames.

00:12 So we have a frame of daniela, the frame of james, and the frame of terra.

00:17 And we're asked to find, given the values of the ratios of the times measured of james, with respect to daniela and terra, what is the time? measured by terra as a function or as a ratio of the time measured by daniela.

00:39 So even just saying that out loud sounds pretty complicated.

00:43 But what's actually happening is not too bad.

00:45 So what we have is we have a stationary frame where james is sitting.

00:50 And so as the spaceship of daniela moves past james, since they have synchronized clocks, you know, they would expect their clocks to, their cluster measuring the same time at a given point in the past.

01:05 And so starting from the same point, daniela and terra have accelerated to some values of some velocities, which are probably some fractions of the speed of light.

01:16 And james was staying stationary.

01:18 So, you know, the same time has elapsed, or the appropriate time has elapsed in each of their frames.

01:24 And so what we're told is that the ratio of deniala, time with respect to james's time is 80%, or td over tj is 0 .80.

01:37 And tarras time, with reference to james ' time, is 70%, so it's 0 .70 is the ratio.

01:48 So essentially, if, let's say, you know, james measures 100 seconds, then daniela's clock only actually reads 80 seconds while tarras reads only 70 seconds.

01:59 So what we first have to find is the velocities of tara and daniela in the reference frame of james.

02:08 So the way to do that is to realize that, you know, james is going to be measuring a fraction or is going to be measuring the largest time of all of them because he's staying stationary.

02:19 And so he's going to be measuring the dilated time of the proper times elapsed in the two or on the two spaceships.

02:27 So james ' time is the dilated time.

02:31 So t -prime stands for either daniele's or terra's time, and gamma is the appropriate factor with either danielas or terra's speed.

02:44 So rearranging this, we find that the ratio of t -prime over t -j is equal to 1 over gamma, which is just the square root of 1 minus v squared over c -squared.

02:54 So we can rearrange this general formula for v.

02:59 And if we do that, we first square both sides.

03:01 So we get t prime or t j squared is equal to 1 minus v squared over c squared.

03:05 Then we move v squared over c squared to the left and t prime over t j squared through the right...

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