The positive muon (µ+), an unstable particle, lives on average 2.20 * 10^-6 s (measured in its own frame of reference) before decaying. (b) What average distance, measured in the laboratory, does the particle move before decaying?

Video transcript

Hello, fellow physicists today, we're going to solve the following physics practice problem together. So first off, let's read the pro and highlight all the key pieces of information that we need to use in order to solve this problem. So the question is highly energetic particles originating from cosmic rays approach the surface of earth with a speed close to the speed of light. The so or Kon reaches the earth's atmosphere with a speed of 0.88 C. The mean lifetime of K meson at rest is 1.23 multiplied by 10 to the power of negative eight seconds. Calculate the average distance as measured from earth's frame. The K Meon particles travel before decay. OK. So our end goal is to calculate the average distance as measured from earth's frame. Earth's frame of reference, the K meson particles travel before the K Awesome. So we're given some multiple choice answers and they're all in the same units of M for meters. So let's read off the multiple choice answers to see what our final answer might be. A is 3.25 B is 3.69 C is 6.84 and D is 7.77. So first off, let's recall the time dilation equation since we need this equation in order to solve and or I should say calculate the average distance. So the time dilation equation states that delta T equals delta T subscript zero divided by the square root of one minus B divided by C squared where V is the speed and C is the speed of light. And the numerical value for the speed of light is 3.0 multiplied by 10 to the eighth power meters per second. Awesome. We also need to recall the distance traveled equation which states that D equals V the speed multiplied by delta T. So we also need to note, let's make a little side note here that the lifetime measured at rest for K Nissan is the proper time. So since it's a proper time, it would be delta T zero equals 1.23 multiplied by to the negative eight seconds. OK. So this is for the lifetime measured at rest for K Nissan. So it's the proper time. So it's gonna be delta T zero and delta T in this case equals the earth's frame of reference. I'm gonna call it earth's frame. Awesome. We also need to note that the particles are moving with a speed V with respect to the earth's frame of reference. Therefore, the particles will live longer within the earth's frame of reference. OK. So using the time dilation equation, let's solve for delta T by plugging in our known variables. So let's do that together. So delta T equals, and as we determine delta T subscript zero was 1.23 multiplied by 10 to the negative eighth power seconds divided by the square root of one minus. And V. As given to us in the problem was 0.88 C 0.88 C divided by C squared. So note that the CS cancel, which C represents the speed of light. And when you plug that into a calculator, the numerical value you should get is 2.59 multiplied by 10 to the negative eighth power seconds. So now we can use the distance traveled equation. So let's plug in our known variables to solve for D. So D equals and V as we said was zero point 88 multiplied by C which the numerical value for C is zero point. Sorry, let me rephrase. So the numerical value for the speed of light is 3. multiplied by 10 to the eighth power meters per second multiplied by delta T which we determined when we found out earlier that it was 2.59 multiplied by 10 to the negative eighth power seconds. So note that the seconds cancel leaving us with meters, which is what we need the distance to be in meters. So when you plug that into a calculator you should get 6. m, which is our final answer. So that means that c is our correct answer. 6.84 m. Thank you so much for watching. Hopefully that was helpful and I can't wait to see you in the next video. Thanks for watching. Bye.