Hyperbolic Motion: SR Homework Solutions

In summary, we are asked to consider a particle in one-dimensional hyperbolic motion given by x(t)=\sqrt{b^{2}+t^{2}} where b is a constant. We then proceed to find the proper time \tau(t) by taking the derivative of x(t) and using the equation d\tau^2 = dt^2 - dx^2. We also find x and v_x as functions of \tau and use the rotational Lorentz transformation for hyperbolic motion to find the 4-velocity u^{\mu}.
  • #1
Kiwithepike
16
0

Homework Statement


Consider a particle in one-dimensional so called hyperbolic motion
x(t)=[itex]\sqrt{b^{2}+t^{2}}[/itex]
where b is a constant.

a) Find[itex]\gamma[/itex](t).
b) Find the proper time [itex]\tau[/itex](t). (assume that [itex]\tau[/itex]=0 when t = 0
c) Find x and v[itex]_x[/itex] as functions of the propertime [itex]\tau[/itex].
d) FInd the 4-velocity u[itex]^{\mu}[/itex].

The Attempt at a Solution



A) ok to begin I took the derivative of x(t) to get velocity. tuned out to be t(b[itex]^{2}[/itex]+t[itex]^{2}[/itex])[itex]^{-1/2}[/itex].
soo therefor [itex]\gamma[/itex](t) = [itex]\frac{[itex]\sqrt{b^{2}+t^{2}}[/itex]}{[itex]\sqrt{1-\frac{t^{2}}{\sqrt{b^{2}+t^{2}}}}[/itex]}[/itex]

b) so now [itex]\tau[/itex](0) = [itex]\sqrt{t^{2}-(b^{2}+t^{2}}[/itex]
[itex]\tau[/itex](0) = [itex]\sqrt{0^{2}-(b^{2}+t^{0}}[/itex] = 0
[itex]\tau[/itex](0) = [itex]\sqrt{-b^{2}}[/itex] = 0
so would b = 0?
this is where I'm getting lost.
c) x as a function of \tau would be [itex]\sqrt{t^{2}-\tau^{2}}[/itex]=x?
where does v[itex]_x[/itex] come in? would i solve v(t) for t^2?

d) I know the 4 vector for u[itex]^{\mu}[/itex] is (u^0,u^1,u^2,u^3) and the roattional lorrentz for hyperbolic is
|t'| = |cosh[itex]\varphi[/itex] -sinh[itex]\varphi[/itex] |
|x'| |-sinh[itex]\varphi[/itex] cosh[itex]\varphi[/itex] |

where tanh[itex]\varphi[/itex]=v
where cosh[itex]\varphi[/itex]= [itex]\gamma[/itex]

where do i go from here? Thanks for all the help.
 
Last edited:
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  • #2
Kiwithepike said:
b) Find the proper time [itex]\tau[/itex](t). (assume that [itex]\tau[/itex]=0 when t = 0

Try using ##d\tau^2 = dt^2 - dx^2##.
 

Related to Hyperbolic Motion: SR Homework Solutions

1. What is hyperbolic motion?

Hyperbolic motion is a type of motion in which an object moves in a curved path known as a hyperbola. This type of motion is governed by the laws of special relativity and is commonly observed in high-speed or near-light-speed situations.

2. How does special relativity apply to hyperbolic motion?

Special relativity is a theory that explains the behavior of objects moving at high speeds or in accelerated frames of reference. It is used to calculate the effects of time dilation and length contraction, which are crucial factors in understanding hyperbolic motion.

3. What are some real-life examples of hyperbolic motion?

Some examples of hyperbolic motion include the motion of particles in a particle accelerator, the orbit of comets around the sun, and the motion of stars in a binary star system. These situations involve objects moving at high speeds and accelerating forces, making them ideal scenarios for studying hyperbolic motion.

4. What is the equation for calculating hyperbolic motion?

The equation for calculating hyperbolic motion is known as the hyperbolic trajectory equation, and it is given by x(t) = x0 + v0t + (1/2)at^2 + (1/2)(γ-1)a(v0/c)^2t^2, where x0 is the initial position, v0 is the initial velocity, a is the acceleration, and γ is the Lorentz factor.

5. How is hyperbolic motion different from other types of motion?

Hyperbolic motion is different from other types of motion, such as linear or circular motion, because it involves objects moving at near-light speeds. This leads to unique effects, such as time dilation and length contraction, that are not observed in other types of motion.

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