A spaceship of mass m is traveling away from Earth at speed...

A spaceship of mass $m$ is traveling away from Earth at speed $v$. Its momentum has magnitude $2.5 \mathrm{mv}$. (a) Find $v$.
(b) An astronaut on the spaceship has a watch that ticks once every second. How often does the watch tick as measured by an Earth observer?

Key Concepts

Relativistic Momentum

Relativistic momentum generalizes the classical concept of momentum to account for the effects of high velocities approaching the speed of light. It is defined as p = ?mv, where ? (the Lorentz factor) modifies the mass and velocity relationship to include relativistic corrections, distinguishing it from the Newtonian formulation.

Lorentz Factor

The Lorentz factor, given by ? = 1/?(1 ? v²/c²), emerges from Einstein's theory of special relativity and governs how time, length, and mass behave at high speeds. It quantifies the degree of relativistic effects, becoming significantly greater than 1 as the speed v approaches the speed of light c.

Time Dilation

Time dilation is a fundamental phenomenon in special relativity where a clock moving relative to an observer ticks slower than a clock at rest with respect to that observer. This effect is directly related to the Lorentz factor, so that the proper time measured in the moving frame is shorter than the time interval measured in the observer's frame.

Transcript

00:01 Okay, so today we're going to take a look at problem number 37 here.

00:06 So here we have an object, a spaceship in particular.

00:10 It's got a certain mass m and it's moving away from the earth at a certain speed v.

00:16 And we're told that its momentum has a magnitude that is 2 .5 times the newtonian approximation of the momentum, which is mv.

00:25 And first of all we want to find the v and then and then we have the part b where we're comparing the like we're talking about the time dilation effect in part b so let's do part a first so here for part a we want to calculate the the v the speed here so let me just write down the expression for the relativistic momentum which is mv almost mv but not quite because there's a gamma factor, which is one over, one over the square root of one minus v squared over c squared.

01:03 So this is the relativistic version of the momentum, and we're told that this is 2 .5 times mv.

01:11 Right, so there's an mv here.

01:14 There's an mv here.

01:15 They cancel.

01:16 So here we just have one over the square root of one minus v squared over c squared.

01:23 That is approximately, that is exactly 5 over 2.

01:27 So now if you solve this equation here for the variable v, you're going to find that v is approximately 0 .9 times the speed of light.

01:39 It is very close to the speed of light.

01:45 So this is the answer for part a.

01:48 It's fairly straightforward.

01:50 It just involves this solving of this equation here.

01:55 Now let's do part b.

01:58 Part b is basically where you have an astronaut on the spaceship, and according to the astronaut, the watch ticks once every second.

02:07 And we all know that one second for the astronaut is not one second for someone on earth, because there's relative velocity between the astronaut and person on earth.

02:19 So with a relative velocity comes time dilation.

02:24 So what the astronaut perceives as one second is not one second for someone on earth.

02:31 And we have this expression here, neat little expression here for the time dilation.

02:38 We have d -tow is d t times the square root of the 1 minus v squared over c squared, which is basically like the gamma factor, but the inverse of that...

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