Help with relativity and time dilation

[jaterp]

This problem i got for school has been driving me crazy >.<

"As seen by one observer, two firecrackers are set off at the same place but separated by 4 seconds in time. As observerd by a second observer, moving with respect to the first, the two firecrackers detonate at places separated by 3 light-seconds in space. As observed by this second observer, how much time is there between the two detonations?
How fast is this second observer moving with respect to the first?"

So basically for the second observer, he sees the detonation happen at different places coz he's moving right ? but where am i suppose to go from there ? I solved for v with time dilation thinking that second observer sees the time as 3 seconds but when i re-read the question i realized i totally misunderstood it and that it was distance between the detonation. Any sort of help will be really appreciated ^^

 

[Doc Al]

Start by writing the Lorentz transformations that relate distance and time measurements made in different frames.

 

[quicknote]

Hello there,

I can understand how this problem may be causing you frustration. The concept of relativity and time dilation can be quite complex and difficult to wrap our minds around. However, with some explanation and guidance, I'm sure you will be able to understand and solve this problem.

Let's start by breaking down the problem and understanding the key concepts involved. The first concept we need to understand is that of relativity. This is the idea that the laws of physics are the same for all observers, regardless of their relative motion. This means that if two observers are moving at different speeds, they will still observe the same physical laws and phenomena.

Now, let's look at the scenario presented in the problem. We have two observers, one stationary and one moving, observing two firecrackers being set off at the same place but at different times. The key here is that the two observers are moving at different speeds, which means they will experience time differently. This is where time dilation comes in.

Time dilation is a concept in relativity that states that time slows down for an observer who is moving at high speeds. This means that the moving observer will perceive time passing slower than the stationary observer. In this case, the moving observer will see the two firecrackers detonating at different times, even though they were set off at the same time for the stationary observer.

Now, let's move on to the actual problem. The second observer sees the two firecrackers detonating at different places because of their relative motion. The distance between the detonations is 3 light-seconds, which means that the moving observer is traveling at a speed of 3 light-seconds per second. To find the actual speed, we need to convert this to a more familiar unit, such as meters per second.

To do this, we can use the formula for time dilation, which states that the observed time is equal to the actual time multiplied by the square root of 1 minus the square of the observer's velocity divided by the speed of light squared. In this case, we know the observed time (3 seconds) and the actual time (4 seconds). We also know that the speed of light is approximately 299,792,458 meters per second.

Using this information, we can solve for the velocity of the moving observer. When we plug in the values and solve for v, we get a velocity of approximately 224,844,343 meters per second. This is the speed