Relativistic addition of velocities problem

AI Thread Summary
The discussion revolves around calculating the relative speed of a truck approaching a car using the relativistic addition of velocities formula. For part A, the initial calculations were incorrect due to misinterpretation of the velocities relative to the speed of light, leading to a corrected relative speed of 49 m/s. Participants clarified that even at lower speeds, such as 5 m/s, the relativistic equation is preferable unless both speeds are significantly smaller than the speed of light. The conversation emphasizes the importance of correctly applying the relativistic formula to achieve accurate results in velocity calculations. Understanding these principles is crucial for solving similar physics problems effectively.
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Homework Statement



You are driving down a two-lane highway and a truck in the opposite lane travels toward you. Suppose the speed of light in a vacuum is 65 m/s. Determine the speed of the truck relative to you when

a - your speed is 25 m/s and the truck's speed is 35 m/s and

b - your speed is 5 m/s and the truck's speed is 55 m/s

The speeds given are relative to the ground

Homework Equations



V(ab) = Vac + Vcb / 1 + (Vac*Vcb/c^2)

The Attempt at a Solution



The velocity of the car (me) relative to the ground is Vac or 25 or 0.25c
Velocity of truck in opposite direction is Vcb or -35 or -0.35c

Plugging into relativistic addition of velocities:

0.25c - 0.35c / 1 - (0.0.875)= -0.12c
Given that c = 65 for this problem, Vab = (-0.12)(65) = -7.1 m/s

I'm pretty sure that answer is wrong, because it makes no sense that the truck would be traveling at 7m/s relative to the car but I cannot figure out what I'm doing wrong here.

And for part B, I'm a little unclear on how "close" to the speed of light the velocities have to be in order to use the relativistic equation. Is 5 m/s far enough from the given speed of light (65 m/s) that I should use the "original recipe" addition of velocities to find the velocity of the truck? How would I know this?

Any help is greatly appreciated!
 
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123456 said:
The velocity of the car (me) relative to the ground is Vac or 25 or 0.25c
Va/c = 25 m/s (not 0.25c); (Va/c is the speed of you "a" relative to the ground "c").

Velocity of truck in opposite direction is Vcb or -35 or -0.35c
Careful, Vc/b is the velocity of the ground with respect to the truck ("b"), thus Vc/b = + 35 m/s.

The speed of light is 65 m/s. Va/c = 25 m/s ≠ .25c
 
Oh! So, Vac = 25/65 = 0.38c and Vcb = 35/65 = 0.58c

So, plugging in:

0.38c+0.54c / 1 + (0.38c)(0.54c)/c^2
= 0.71c, so truck is traveling 49m/s relative to the car? That makes more sense, because it would seem to be traveling faster than it actually is when viewed from the car's frame. Is that right?

For Part B, can I use the relativistic equation or is 5 m/s not close enough to the speed of light?
 
123456 said:
Oh! So, Vac = 25/65 = 0.38c and Vcb = 35/65 = 0.58c

So, plugging in:

0.38c+0.54c / 1 + (0.38c)(0.54c)/c^2
= 0.71c, so truck is traveling 49m/s relative to the car? That makes more sense, because it would seem to be traveling faster than it actually is when viewed from the car's frame. Is that right?
Yes. The speed of the truck in the frame of the car is greater than the speed of the truck in the frame of the Earth (since the car is also moving with respect to the earth).

For Part B, can I use the relativistic equation or is 5 m/s not close enough to the speed of light?
Unless both speeds are much smaller than the speed of light, it's best to use the relativistic formula. (For small speeds, the relativistic formula gives almost the same results as the galilean formula.)
 
OK, I think I understand a little more. Thanks so much for helping me!
 

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