An alien spacecraft is flying overhead at a great distance as you stand in your backyard. You see its searchlight blink on for 0.150 s. The first officer on the spacecraft measures that the searchlight is on for 12.0 ms. (a) Which of these two measured times is the proper time? (b) What is the speed of the spacecraft relative to the earth, expressed as a fraction of the speed of light c?

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. So a control tower operator detects a red light pulse emitted from an unidentified flying object UFO in earth's atmosphere at a speed of V. The control tower operator measures the pulse duration to be 1. seconds. A passenger on the UFO observes the light pulse lasts for 1.1 seconds. I who measures the proper time interval for the duration of the pulse. I, I express the speed V of the UFO in the earth's frame as a function of the speed of light C. Awesome. So we have two goals that we're trying to accomplish here. So the first one is that is to determine who measures the proper time interval for the duration of the pulse. And the second goal is to express the speed V of the UFO in the earth's frame as a function of the speed of light C. OK. So we're given some multiple choice answers. Let's read them off to see what our final answer might be. A is I, the control tower operator measures the proper time I I V equals 0.53 C B. Is I the control tower operator measures the proper time I I V equals 0.63 cc is I the UFO passenger measures the proper time I I V equals 0.53 C and D is I the UFO passenger measures the proper time I I V equals 0.63 C. Awesome. Awesome. Awesome. OK. So first off to sol for I, we must recall that proper time is determined by the frame of reference in which both events take place. In this case, the light emission. So the red light pulse and the observation of the poles, which in this case, both of them take place at the same time. So therefore the passenger on the UFO measures the proper time. So for I, the UFO measures the proper time. Awesome. So to solve for I I, we need to express V as a function of C. So to do that, we need to use the time dilation equation and rearrange it to solve for B. So let's recall that the time dilation equation is as follows. So delta T equals delta T subscript zero divided by the square root of one minus V divided by C squared. OK. So we also need to make a quick little note here that in this case, delta T subscript zero is the proper time. So that's the UFO observation which as defined as given to us in the equation that is 1.1 seconds. So the time duration of the pulse observed by the UFO is 1.1 seconds. And delta T which was the control tower observation of the pulse was said to be 1.3 seconds is the duration. OK. Awesome. So we need to use algebra to rearrange the time dilation equation to solve for B. So we need to isolate V. So when we do that, we'll skip ahead here. So when you do all the rearranging using algebra, when you get V by itself, you should get V equals C square root of one minus delta T subscript zero divided by delta T squared. Awesome. So now at this stage, we can plug in our known variables to solve for C. So let's do that. So V equals C multiplied by the square root of one minus 1.1 divided by 1.3 squared. And when you plug that into a calculator, you should get 0.533 C. So that means our final answer must be C I, the UFO passenger measures the proper time I I V equals 0.53 C. Thank you so much for watching. Hopefully, that helped and I can't wait to see you in the next video. Bye.