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Problem 53 Hard Difficulty

An alien spaceship traveling at $0.600 c$ toward the Earth launches a landing craft. The landing craft travels in the same direction with a speed of $0.800 c$ relative to the mother ship. As measured on the Earth, the spaceship is 0.200 ly from the Earth when the landing craft is launched. (a) What speed do the Earth-based observers measure for the approaching landing craft? (b) What is the distance to the Earth at the moment of the landing craft's launch as measured by the aliens? (c) What travel time is required for the landing craft to reach the Earth as measured by the aliens on the mother ship? (d) If the landing craft has a mass of $4.00 \times 10^{5} \mathrm{kg}$, what is its kinetic energy as measured in the Earth reference frame?

Answer

$0.946 c$

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Video Transcript

So here in this problem, we are given that speed of the alien ship. With respect to the Earth, we have given us 0.6 c and the speed of the Landau. With respect to the mothership, we are given us you x dash us 0.8 c. Now we'll be taking the earth as R s free and the alien ship as R s dash frame. And there are four parts in the question will be solving the apart first. Now, in a part, we have to calculate the speed of the lender with respect to the earth. So let it be you, x, And by using the Lorentz transformation equation for velocity the speed of Landau with respect to Earth, you x will be your ex prime or you X dash plus V divide by one plus you X dash V over C squared. Now we have the village. You can plug it here, you x dash. We have 0.8 c plus. We we have 0.6 c divided by one plus you x dash. We have 0.8 c. Times V is 0.6 c. Divide by C square. No goodness all this and we'll get the answer. You excess zero point 94 60. So this dancer for the a part next will be solving the B part. Now, in the B part, we have to calculate additions to the oath. At the moment the landing craft was launched as measured by the aliens. So and we are given that the distance between the lender and the Earth, as observed by the earth, uh, is LP were given us 0.2 little So l is equal to l P times underwrote one minus the square over C square, so it would be equal to L. P. We have 0.2 light years. Times underwrote one minus. The square means 0.6 square, 0.6 c square. Sorry. Divide by C Square as it is. No goodness, All this and we'll get our answer US 0.16 light here. So there's the answer for the B part now. Next will be solving, though. See part now in the C part. We have to calculate the time travel or the time required for the landing craft to reach the Earth as measured by the aliens so basically we have to calculate so t would be called to distance divided by the speed of the lender. Not just the distance is 0.16 light here, 0.16 light here that was major by the aliens divided by the speed and speed would be 0.8 C plus 0.6 c Not going to solve this, and we'll get our answer us 0.11 for you. So this answer for the C part next it will be solving the deep part now in the report were to calculate the kinetic energy as measured in the Earth's reference frame, and the expression for it is given by Canada. Calls to one over on the Route one minus U X square over C square minus one times M C squared. Now we have all the valleys. We can plug it here, and this is the mosque of the Lindo, which we are given us MM recalls to four times 10 to the power five kg. This is given the question. Now I can put the values so one over underwrote one minus U x squared that we have calculate earlier as 0.946 See whole square divided by C square minus one times. And we have four times 10 to the power five and the value of C. We have three times 10 to the power eight media per second. This is the speed of light. I'm going to simplify this and we'll get our answer as 7.5 times 10 to the power 22 June. So this is required Canada Energy. That was asking the question. So I hope you have understood the problem. Thank you.

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