The positive muon (µ+), an unstable particle, lives on average 2.20 * 10^-6 s (measured in its own frame of reference) before decaying. (a) If such a particle is moving, with respect to the laboratory, with a speed of 0.900c, what average lifetime is measured in the laboratory?

Video transcript

Hello, fellow physicists today, we're to solve the following practice problem together. So first off, let's read the problem and highlight all the key pieces of information that we need to use. In order to solve this problem. A high energy particle accelerator produces unstable particles moving along the horizontal axis at a speed of 0.810 C in the laboratory frame at rest these particles decay with an average lifetime of 0.80 microseconds to calculate the lifetime of the particles measured with the detector fixed in the laboratory. Awesome. OK. So our goal is to calculate the lifetime of the particles measured with the detector fixed in the laboratory. OK. Awesome. So we're given some multiple choice answers and they're all in the same units of microseconds. So let's read them off to see what our final answer might be. A is 0.47. B is 0.80 C is 1.36 and D is 2.31. Awesome. So first off, let's recall and use the time dilation equation. So the time dilation equation states that delta T is equal to delta T subscript zero divided by the square root of one minus B divided by C squared. So in this case, delta T is going to represent the lifetime of the particles measured with a detector fixed in the laboratory. So that's the value that we're trying to solve for is delta T and delta. Here, we'll make a side note in delta T subscript zero is the decay lifetime of particles at rest, which in the prot defines it to be 0.80 microseconds and C is speed of light and V is the speed of the particles with respect to the live laboratory frame. OK. So to solve for the lifetime of particles measured with a detector fixed in the laboratory, we must solve for delta T. So let's plug in our known variables. So, oh wait, yeah. And let's note that V in this case, the speed is equal to 0.810 C. OK. So delta T zero or delta T subscript zero is 0. micro seconds divided by the square root of one minus. And V in this case is 0.810 C divided by C squared. So note that the CS will cancel out. So the speed of light cancels out. And when you plug this into a calculator, you should get 1. microseconds. Awesome. So that's our final answer. That one was a quick problem. So our final answer is C 1. microseconds. Awesome. Thank you so much for working with me. I can't wait to see you in the next video. Hopefully this helped and thank you for watching. Bye.