What is the velocity of car 2 when car 1 is going 1/2 the speed of light?

In summary, car 2 is twice as long as car 1 when they are at rest, but when car 2 is moving at a speed of sqrt(13/16)c, the length of car 2 is equal to the length of car 1 as observed by a stationary policeman in the same reference frame. The equation L 1/sqrt(1-(1/2)^2) = 2L/sqrt(1-v^2/c^2) is used to calculate the length of car 1 and car 2 in this scenario, but the sqrt(1-v^2/c^2) should be in the numerator on both sides to accurately reflect the length contraction effect at high speeds.
  • #1
ehrenfest
2,020
1
car 2 is twice as long as car 1 when they are at rest

a stationary policeman observes that car 2 is the same length as car 1 as car 2 passes car 1 going through a speed trap.

car 1 is going 1/2 the speed of light

I tried to solve this with

L 1/sqrt(1-(1/2)^2) = 2L/sqrt(1-v^2/c^2) and I got an imaginary anwer
 
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  • #2
ehrenfest said:
car 2 is twice as long as car 1 when they are at rest

a stationary policeman observes that car 2 is the same length as car 1 as car 2 passes car 1 going through a speed trap.

car 1 is going 1/2 the speed of light

I tried to solve this with

L 1/sqrt(1-(1/2)^2) = 2L/sqrt(1-v^2/c^2) and I got an imaginary anwer

your equation is wrong. the sqrt(1-v^2/c^2) should be in the numerator on the RHS

[edit: and the LHS too]
 
Last edited:
  • #3
Why? I took the stationary policeman as my intertial reference frame. To him L 1/sqrt(1-(1/2)^2) is the length of car 1 and 2L/sqrt(1-v^2/c^2) is the length of car 2. These quantities must be equal, right?
 
  • #4
ehrenfest said:
Why? I took the stationary policeman as my intertial reference frame. To him L 1/sqrt(1-(1/2)^2) is the length of car 1 and 2L/sqrt(1-v^2/c^2) is the length of car 2. These quantities must be equal, right?

As olgran said that sqrt should be in the numerator on both sides. The length contracts... Can you explain the steps you took to get it in the denominator?
 
  • #5
ehrenfest said:
Why? I took the stationary policeman as my intertial reference frame. To him L 1/sqrt(1-(1/2)^2) is the length of car 1 and 2L/sqrt(1-v^2/c^2) is the length of car 2. These quantities must be equal, right?

Nope. For example, to him the length of car one is

[tex]
\frac{L}{\gamma(c/2)}\;.
[/tex]

Which is *not* the same as what you have written.

Don't get your gammas upsidedown.
 
  • #6
remember length contracts when moving and proper length is always biggest. and since [tex]\gamma \geq 1[/tex] always, you can quickly check whether your answer make sense or not... for [tex]\frac{L}{\gamma} \leq L[/tex]
 
  • #7
I see. So the velocity of car 2 must be sqrt(13/16)c, right?
 
  • #8
ehrenfest said:
I see. So the velocity of car 2 must be sqrt(13/16)c, right?

yep.
 

Related to What is the velocity of car 2 when car 1 is going 1/2 the speed of light?

1. What is length contraction of cars?

Length contraction of cars is a physical phenomenon predicted by Einstein's theory of special relativity. It states that as an object moves at high speeds, its length in the direction of motion will appear to contract when observed by an outside observer. This means that a car traveling at high speeds will appear shorter to someone watching it from a stationary position.

2. How does length contraction of cars occur?

Length contraction occurs because of the concept of time dilation. As an object moves at high speeds, time appears to slow down for that object. This means that the time it takes for the front of the car to reach a certain point will be longer than the time it takes for the back of the car to reach the same point. This difference in time causes the front of the car to appear closer to the back, resulting in the appearance of length contraction.

3. Does length contraction of cars only occur at very high speeds?

Yes, length contraction of cars is only noticeable at speeds close to the speed of light. At everyday speeds, the difference in length would be too small to be noticeable to the human eye. However, it is still a real physical phenomenon that occurs at all speeds, just not noticeable at lower speeds.

4. What are the implications of length contraction of cars?

The implications of length contraction of cars are significant in the field of physics and transportation. It means that the length of a car, and any other object traveling at high speeds, is not a fixed value and can change depending on the perspective of the observer. This concept is crucial in understanding the behavior of objects at high speeds and has implications for space travel and other advanced technologies.

5. Can length contraction of cars be observed in daily life?

No, length contraction of cars is not something that can be observed in daily life. As mentioned earlier, it only becomes noticeable at speeds close to the speed of light, which is not achievable by cars or any other object in our everyday lives. However, it can be observed and measured in particle accelerators, where particles are accelerated to very high speeds and their length contraction can be observed and calculated.

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