As you pilot your space utility vehicle at a constant speed toward the moon, a race pilot flies past you in her spaceracer at a constant speed of 0.800c relative to you. At the instant the spaceracer passes you, both of you start timers at zero. (a) At the instant when you measure that the spaceracer has traveled 1.20 * 108 m past you, what does the race pilot read on her timer? (b) When the race pilot reads the value calculated in part (a) on her timer, what does she measure to be your distance from her? (c) At the instant when the race pilot reads the value calculated in part (a) on her timer, what do you read on yours?

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Accepted Answer

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. Two synchronized stop watches are placed in two spaceships A and B spaceship A is traveling with a velocity of 0.6 C relative to spaceship B. When spaceship A passes spaceship B, the two stop watches start to count time simultaneously. I determine the time T A T subscript capital A displayed on the stopwatch of spaceship A when spaceship A has moved 0.75 multiplied by 10 to the power of eight m in front of spaceship B I I, what is the distance separating the two spaceships at the instant when the stopwatch in spaceship A displays the time T A and I I I, what time TB does the stopwatch of spaceship B display? When the stopwatch of spaceship A displays the time T A? Awesome. So we have a lot going on. So we have three goals that we our, our main. So to find our final answer, we have to solve three different solve for three different things. And that is our end goal is to solve for each answer. One for I, I, I, and I, I, I, OK, so we're given some multiple choice answers and all the answers for I, for T A are all in seconds and all the answers for I, I, I are in meters for the distance and all the answers for I, I, I, for TB are in seconds. So let's read them off to see what our final set of answers might be. So for a, it's 0.3340 point 750, multiplied by 10 to the power of seven and 0.417. B is 0.3346 point 01 multiplied by 10 to the power of seven and 0.417. C is 0.4170 point 750, multiplied by 10 to the power of seven and 0.334. And finally, for D 0.1, sorry, 0.4176 point 01 multiplied by 10 to the power of seven and 0.334. OK. So first off, we need to recall and use the time dilation equation which states that delta T equals delta T subscript zero, delta T zero, divided by the square root of one minus V divided by C sorry, one minus V squared divided by C square. OK. Where T zero is the time displayed on stop wa displayed on the stopwatch on spaceship B and delta T zero is the proper time displayed by the stopwatch on spaceship A. OK. So we also need to recall how to solve for delta T, which to do that. We need to recall that to find Delt all it is is the distance as the distance traveled divided by the speed of the station or speed in this case. Well, in this case, it's the speed of the spaceship. So it's distance travel divided by the speed. So let's plug in our known variables to solve for T delta T. So the distance traveled by spaceship A as given to us at the beginning of the, in the problem, it's 0.75 multiplied by 10 to the power of eight meters. And then the speed of the spaceship A with respect to spa with spaceship B has given us to the in the problem as 0.6 C which let's make a quick little note here that C equals the speed of light. And the numerical value for that is 3.0 multiplied by 10 to the power of eight meters per second. So that's let's plug in C. So the distance travels at 0.75 multiplied by 10 to the power of eight m divided by 0.6 multiplied by 3.0, multiplied by 10 to the power of eight m per second. So when you plug that into a calculator, you should get 0. seconds. Awesome. So we also need to make a quick, a couple of quick little notes here. We need to note that the time for T A, so the time for spaceship A equals delta T zero and the time for spaceship BT BT subscript B equals delta T. OK. Which means if T BT substitute equals delta T, then that means our answer for I I has to be, I mean, I should, I should be clear for TB which would be for I I I. So this is the answer for I I I. So we already found one answer hooray, but we need this to solve for our delta T zero. So let's do that. OK? So now we can use our time dilation equation to solve for delta T zero. But first off before we can solve for it, we need to quickly rearrange it to solve for delta T zero. So when we rearrange it to solve for delta T zero, the equate for time dilation equation should look like this. Now solving for T delta T 01 minus B divided by C squared multiplied by delta T. So let's plug in our known variables and sol for delta T zero, it's one minus and then it was the velocity or the speed is 0.6 C squared divided by C squared note that the CS the speed of light cancels out. And that's multiplied by delta T, which we determine to be 0.417 seconds. So delta T zero equals when you plug this into a calculator, you should get 0.334 seconds. And this is the answer for I Awesome. So now we can start solving for I, I, so let's do that. So to solve for I I, we need to recall the equation for distance traveled. So the equation for distance travel, which we're trying to find the distance traveled for spaceship A. So we're gonna call it D subscript capital A equals the velocity multiplied by delta T. So let's plug in our known variables to solve for the distance traveled by spaceship A. OK. So we know that our velocity was 0. C 0.6. And we need to multiply it by C which is value for C L A is 3.0 multiplied by 10 to the power bait meters per second. And then we determine the value for delta. Oh, it should be delta T zero. My bad K multiplied by T zero was 0. seconds. Awesome. OK. So when you plug that into a calculator, our distance traveled for spaceship A should equal 6.1 multiplied by 10 to the power of seven m means the seconds will cancel out just leaving meters. Ok. So that means that our answer for I I is 6.1 multiplied by 10 to the seventh power. So that means we solved for everything we needed to solve for our hurray. So that means that our final answer must be B I equals T subscript capital A. So T A equals 0.334, I, I equals D equals 6.1 multiplied by 10 to the power of seven m. And I I I TB equals 0.417 seconds. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

Verified Solution

Key Concepts

Relativity of Simultaneity

Time Dilation

Lorentz Transformation