An unstable particle is created in the upper atmosphere from a cosmic ray and travels straight down toward the surface of the earth with a speed of 0.99540c relative to the earth. A scientist at rest on the earth’s surface measures that the particle is created at an altitude of 45.0 km. (a) As measured by the scientist, how much time does it take the particle to travel the 45.0 km to the surface of the earth? (b) Use the length-contraction formula to calculate the distance from where the particle is created to the surface of the earth as measured in the particle’s frame. (c) In the particle’s frame, how much time does it take the particle to travel from where it is created to the surface of the earth? Calculate this time both by the time dilation formula and from the distance calculated in part (b). Do the two results agree?

Video transcript

Hello, fellow physicists today, we're gonna 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 radioactive nuclei at a high altitude H equals 65.0 kilometers emits a particle that travels at 0. C toward the surface of a planet. Both H and speed are measured by a stationary observer on the planet's surface. I determine the time taken to cover H as measured by the observer on earth. I I use relativity to determine the value of H as measured in the particle's inertial frame. II, I use relativity to determine the time the particle takes measured relative to its inertial frame. OK. So our end goal is to find three separate answers. Our first answer we're trying to find is to find the time taken to cover H as measured by the observer on earth. Our second answer is we need to use relativity to determine the value of H as measured in the particle's inertial frame. And then the last answer we're trying to find we need to use relativity to determine the time the particle takes measured relative to its inertial frame. OK. So we're given some multiple choice answers here. So let's read them off to see what our final answer pair might be. A is I is 4.11 multiplied by 10 to the power of negative seven seconds. I, I is 65.0 kilometers II, I is 4.11 multiplied by 10 to the power of negative seven seconds. B is I 4.11 multiplied by 10 to the power of negative four seconds. 55.2 kilometers for I I. And then II, I is 3.49 multiplied by 10 to the power of negative four seconds. C is one 123 seconds for I and 65.0 m for I I and then II, I is 104 seconds. D is 1.23 multiplied by 10 to the power of five seconds. I I is 34.2 kilometers and II I is 6.48 multiplied by 10 to the power of four seconds. OK. So first off, let us make the following note, we need to note that the observer is stationary on the planet's surface. Thus, the observer will make proper measurements of the separation between the nuclei creation point and the planet's surface. So that means that I subscript zero I zero is equal to 65. kilometers. So to solve for part I, we need to determine the time taken to cover H which is the altitude. So to do that, we need to use the equation T is equal to I zero divided by V where V is the velocity. OK. So now let's plug in our known variables to solve for T. So we know that I zero is 65.0 kilometers, but we need to convert kilometers to meters. So all we have to do is just take 65.0 multiply it by 10 to the power of three to convert kilometers to meters. And it's divided by the velocity. And let's note that the velocity is given to us in the problem as 0.52689 C where C represents the speed of light. So we need to multiply it by the speed of light. And let's recall that the speed of light is the numerical value is 3.0 multiplied by 10 to the power of eight meters per second. Awesome. So that means when we plug into a calculator, the value for T that we should get is 4. multiplied by 10 to the power of four seconds. Awesome. So this is the answer her eye. So I equals 4.11 multiplied by 10 to the power of four seconds. OK. So now we could start solving for I I we're on a roll. So we need to recall and use the Special Relativity equation to find the value of H as measured in the particles inertial frame. So note that I equals I zero multiplied by the square root of one minus U squared divided by C squared. So we now need to take this equation and plug in all of our known variables to solve for I. So let's do that. So I equals I zero, which was 65.0 multiplied by 10 to the power of 3 m is kilometers, converts, converted to meters multiplied by the square root of one minus the speed which was 0. 526989. Hm Let's write that a little more neat. Shall we? aN:aN:000NaN 0. C squared divided by C squared note that the C squares cancel out. So when we plug that into a calculator, we should get five point multiplied by 10 to the power of 4 m. But let's convert that to kilometer. So all we have to do is just multiply it by 10 to the power of negative three. So when we do that, we should get 55. kilometers. So I equals 55.2 kilometers. And this is our answer for I I great, we're making a lot of progress here. OK. So now let's solve for part II I so to solve for II I, we need to recall and use the equation for time dilation. So the equation for time dilation states that delta T zero is equal to delta T multiplied by the square root of one minus U squared divided by C squared. OK. So let's plug in all of our known variables to solve for delta T zero. So delta T is equal to 4.11 multiplied by 10 to the power of four seconds, which is the value we found for part I multiplied by the square root of one minus our speed, which was 0. C squared, divided by C squared. Like before C squared cancels out. So when we plug that into a calculator, we should get our value for delta T zero is equal to 3.49 multiplied by 10 to the power of negative four seconds. And this is our answer for II I array, we did it. So let's go back to the top and look at our multiple choice answers to see what the final answer has to be. And it's the letter B I is 4.11 multiplied by 10 to the power of negative four seconds. I I is 55.2 kilometers and II I is 3.49 multiplied by 10 to the power of negative four seconds. Thank you so much for watching. Hopefully, that helps. And I can't wait to see you in the next video. Bye.