WEBVTT
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Suppose the speed of light were only 3000 meters per second.
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A jet fighter moving toward a target on the ground at 800 meters per second shoots bullets, each having a muzzle velocity of 1000 meters per second.
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What are the bulletsβ velocity relative to the target?
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In this problem statement weβre told to imagine that π the speed of light is 3000 meters per second and that a jet fighter is approaching a target on the ground at 800 meters per second, which weβll call π£ sub π, and that the jet fighter fires bullets which leave the jet fighter with a muzzle velocity of 1000 meters per second, which weβll call π£ sub π.
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Knowing all this, we want to solve for the bulletsβ velocity relative to the target, what weβll call π£ sub π‘.
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We can begin by drawing a diagram of our scenario.
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In this scenario, we have our target, the box off to the right, and the jet fighter approaching at a speed π£ sub π and then firing bullets at the target which move relative to the jet with speed π£ sub π.
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We want to know the velocity of the bullets relative to the target, π£ sub π‘.
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And remember, weβre imagining that the speed of light π is just 3000 meters per second.
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To solve for π£ sub π‘, weβll perform a velocity addition where our velocities are high enough that this addition is relativistic; that is, it takes into account the principles of relativity.
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When we add two velocities in this equation, π’ and π’ prime relativistically, that means that instead of simply adding them like we would classically, we now have a denominator, one plus the product of those two velocities weβre adding divided by the speed of light squared.
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If we apply this relationship to our particular scenario, then π£ sub π‘, the speed of the bullets relative to the target, equals π£ sub π plus π£ sub π divided by one plus the product of π£ sub π and π£ sub π divided by π squared.
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Since we know all three of these values, we can plug them in now.
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We substitute 800 meters per second for π£ sub π, 1000 meters per second for π£ sub π, and 3000 meters per second for π.
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When we calculate π£ sub π‘, we find a result of 1.65 kilometers per second.
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This is how fast the bullets would approach the target under relativistic velocity addition.
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Notice that this is slower than if we had just added the velocities of the jet and the bullets classically.