A spaceship shaped like an isosceles triangle has a width of 20.0 m and a length of 50.0 m. Wh

A spaceship shaped like an isosceles triangle has a width of...

Key Concepts

Special Relativity

Special relativity is the physical theory that describes how measurements of space and time differ for observers in different inertial frames of motion. It introduces concepts such as time dilation and length contraction, which become significant at speeds approaching the speed of light. This theory underpins the analysis of how moving objects appear distorted to stationary observers.

Length Contraction

Length contraction is a relativistic effect where the length of an object moving relative to an observer is measured to be shorter along the direction of its motion compared to its proper length (the length measured in the object's own rest frame). This contraction affects dimensions parallel to the direction of motion while leaving perpendicular dimensions unchanged, leading to distortions in the apparent geometry of the object.

Relativistic Transformation of Angles

Due to length contraction affecting only dimensions parallel to the motion, angles in an object’s geometry can appear different when measured in a moving frame compared to the rest frame. In an object like an isosceles triangle, the sides oriented at an angle to the direction of motion will have their effective length altered, resulting in a change in the measured angle between the base and the sides. This concept involves applying trigonometric relations together with the Lorentz contraction factor to determine the observed angle as a function of velocity.

Transcript

00:01 In this problem, we have a frame s and we have a triangular spaceship attached to frame s prime, which is moving to 0 .4c.

00:09 We are given the width of the spaceship and the length of the spaceship.

00:14 Our goal in this problem is to find the angle relative to an observer in s of one side of the ship with the base.

00:23 It's angle theta.

00:24 Now, the ship is at rest in s prime.

00:27 That means these are proper links.

00:31 I'll call it w -0.

00:32 Now call us l0.

00:35 Now what does s, does s see the width contracted? it does not.

00:41 He only get contraction in the direction of motion.

00:44 So l0 is contracted.

00:46 So to s, he will say the width is still w0, but he will give a contracted length l for the length of the ship.

00:55 So part a, we're going to define that angle now.

00:58 So l is equal to l0, square root, 1 minus f squared, over c squared, and tangent theta, which is opposite over adjacent, is going to be l over the w0 over 2, so it's going to be l0, 1 minus v squared over c squared over c squared over 2.

01:27 So this gives me theta, inverse tangent, 50 meters, square root 1 minus 0, 0 ,000, 4c divided by c, square that over 10 meters, 20 divided by 2 is 10.

01:49 And this works out to be 77 .7 degrees.

01:54 That's the angle that s says, side makes with the base.

02:03 Now for part b, we're supposed to do a graph.

02:07 We have our formula here just above, change in theta.

02:15 And let's put in, remember, we're going to be planning this against v over but everything else we want to evaluate.

02:23 So looking here, 50 over 10 is 5.

02:26 No, and the units are gone.

02:29 1 minus v.

02:30 Over c squared.

02:33 That's our formula.

02:35 This is tan theta.

02:38 Now, let's look at some meaning behind this.

02:40 Now, we could be very formal and look at derivatives and look at different points.

02:45 Remember, the derivatives are the slopes of the tangent lines.

02:49 We could look at different points to get a sense of what's happening.

02:53 Overall sense.

02:55 He still would have to be, the way we're going to do it, we're going to be looking at different points here also, so you don't really gain anything.

03:01 And this gives us some more meaning in terms of what these graphs really represent.

03:06 So let's look at the meaning behind this.

03:09 This is the value of tan theta.

03:14 Now remember, v over c will start on at zero.

03:17 So this starts, this first value is five.

03:22 Then it ends up v over c being one, even though we can't reach there.

03:26 Let's just talk mathematics for a second.

03:28 We cannot, this spacecraft cannot reach v equal c, so that we never really reach one.

03:34 So that's just an esotonic value.

03:36 But aseptonically, this becomes, this would be one, and this would be zero.

03:42 So tangent, inverse tangent on zero is zero.

03:47 So this quantity decreases as v over c increases.