How to derive the velocity addition formula

[DODGEVIPER13]

Homework Statement

Derive the formula v= (v'+u)/(1+v'u/c^2) the velcoty addition formula using the below formulas?

Homework Equations

1. vt1=L+ut1
2. (proper time)=(proper length)/v'+(proper length)/c
3. ct2=L-ut2
4. (dilated time)= (proper time)/(sqrt(1-v^2/c^2))
5. L=(proper length)sqrt(1-v^2/c^2)

The Attempt at a Solution

Ok so here is what I did I solved equation 1 above for t1 and got t1=(L+ut1)/v and equation 3 for t2 and got t2=(L-ut2)/c. I then added them together to get (dilated time or delta t)=(L+ut1)/v+(L-ut2)/c. Then I used equation 4 to get ((L+ut1)/v+(L-ut2)/c)sqrt(1-v^2/c^2)=(proper time). Then I set that equal ti equation 2 ((L+ut1)/v+(L-ut2)/c)sqrt(1-v^2/c^2)= (proper length)/v'+(proper length)/c. Then (proper length)=L/sqrt(1-v^2/c^2) and this is where I get lost in my algebra I can't seem to get rid of L and (proper length) in order to find the correct formula?

 

[DODGEVIPER13]

whoops I think I caught some of my errors when I solved for t1 and t2 I forgot the other side?

 

[DODGEVIPER13]

ok after reworking a bit I am still lost but 1 step closer I think. (Proper length)/v'+(proper length)/c=sqrt(1-v^2/c^2)(L/(c+u)+L/(v-u)) I have then tried using the length contraction on this but it gets very complicated uggg.

 

[DODGEVIPER13]

Is it too confusing, should I resubmit

 

[Nickie]

I appreciate your attempt at deriving the velocity addition formula using the given equations. However, there are a few errors in your approach. Firstly, in equation 1, the t1 should be on the left side of the equation, not on the right. Secondly, in equation 3, the L should be on the left side and the ct2 on the right. Thirdly, in equation 4, the square root should be on the entire right side, not just on the (1-v^2/c^2) term.

To derive the velocity addition formula, we can start with the basic equation for relative velocity, which is v = (x2-x1)/(t2-t1). Using this equation, we can find the relative velocity between two frames of reference, say A and B. Let the velocity of frame B with respect to frame A be u and the velocity of frame C with respect to frame B be v'. Then, the velocity of frame C with respect to frame A can be calculated as:

v = (x3-x1)/(t3-t1)

where x3 is the position of an object as observed in frame C, x1 is the position of the same object as observed in frame A, and t3 and t1 are the corresponding times.

Now, we can use the equations given in the homework to express x1 and x3 in terms of x2, t2, and v'. Using equation 1, we have x1 = L + ut1 and using equation 3, we have x3 = L - ut3. Substituting these into the relative velocity equation, we get:

v = (L - ut3 - L - ut1)/(t3 - t1)

Next, we can use the time dilation equation (equation 4) to express t3 and t1 in terms of the proper time (tp) and the time dilation factor (γ) as:

t3 = tp/γ and t1 = tp/γ

Substituting these into the relative velocity equation, we get:

v = (L - u(tp/γ) - L - u(tp/γ))/((tp/γ) - (tp/γ))

Simplifying, we get:

v = (v'+u)/γ

where v' is the velocity of frame C with respect to frame B and u is the velocity of frame B with respect to frame A.

Finally