Please show all work, inlcude units, and cite principles/laws used. Two rockets

Question

Please show all work, inlcude units, and cite principles/laws used.

Two rockets of equal rest length are passing each other head on at relativistic speeds, as shown in the figure below. The black rocket has a gun at its tail, which its gunner fires just as the tail of the white rocket passes the nose of the black rocket. The pilot of the black rocket is furious. He believes the gunner will miss the white rocket completely. The length of the white rocket is contracted, so the projectile will pass in front of the target. The gunner tells the pilot to relax. He points out that in the frame of the white rocket, it is their ship - the black rocket - that is contracted, so the gun is actually aimed at the middle of the white rocket. Does the projectile hit the white rocket or miss it entirely? It cannot do both! Explain where the pilot or the gunner went wrong.

Explanation / Answer

We know that length is contracted if speed is comparable to the speed of light.It does not mean that ship will shrink .it means that length will apear small to the person who is travelling at some non zero speed .The size will remain same but it will apear small to that observer.So to the gunner ,the size of black rocket will look small as he is travelling at some non zero speed but size will be same .So gunner aimed gun by considering that small size but he goes wrong as he thought that black ship will shrink in white ship frame but in white frame ,white ship will be at rest so in that frame ,white ship will shrink as gunner(observer) is in black ship .The length of object (white ship) shrinks .The gunner is wrong and it will not hit the ship

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