An alien spaceship traveling at 0.600 c toward the Earth...

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?

Key Concepts

Relativistic Velocity Addition

This concept refers to the method used in special relativity to combine velocities when objects move at speeds comparable to the speed of light. Unlike classical addition of velocities, the relativistic formula ensures that the resultant speed never exceeds the speed of light and accounts for the effects of time dilation and length contraction when transitioning between different inertial frames.

Length Contraction

Length contraction is a phenomenon in special relativity where the length of an object or the distance between two points, as measured in a moving reference frame, appears shorter than when measured in the object's rest frame. This effect becomes significant as the relative velocity approaches the speed of light, influencing how distances and separations are perceived by observers in different inertial frames.

Time Dilation

Time dilation is a fundamental aspect of special relativity where a clock moving relative to an observer ticks slower compared to a clock at rest with respect to that observer. This effect means that time intervals measured in a moving frame are longer when observed from another frame, which is crucial for calculations involving travel times for objects moving at relativistic speeds.

Relativistic Kinetic Energy

Relativistic kinetic energy extends the classical kinetic energy concept to situations where speeds are close to the speed of light. It accounts for the increase in energy due to the object's mass and its velocity, and uses the Lorentz factor to adjust for the effects of special relativity, ensuring that energy calculations remain accurate in high-speed scenarios.

Transcript

00:01 So here on this problem we are given that speed of the alien ship with respect to the earth we are given as 0 .6c and the speed of the lender with respect to the mother ship we are given us ux dash as 0 .8c now we'll be taking the earth as our s frame and the alien ship as our s dash frame and therefore part in the equation we'll be solving the a part first now in the a part we have to calculate the speed of the lender with respect to the earth so let it be ux and by using the lawrence transformation equation for velocity the speed of lander with respect to earth ux will be ux prime or ux dash plus v divide by 1 plus ux dash v over c square now we have the value you can plug it here ux dash we have 0 .8c plus v we have 0 .6c divide by 1 plus ux dash we have 0 .8c times v is 0 .6c divide by c squared now we'll go to solve this and we'll get the answer ux as 0 .946c so this is the answer part now next we'll be solving the b part now in the b part we have to calculate the distance to the earth at the moment the lending 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 is lp we are given us 0 .2 light air so l is equal to lp times under root 1 minus v square over c square so it would be equals to lp we have 0 .2 light years times under root 1 minus v square means 0 .6 square so 0 .6 c square sorry divide by c square as it is now we'll go to solve this and we'll get our answer as 0 .16 light here so this is the answer for the answer for the b part now next we'll be solving the c 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 t basically we have to calculate so t would be equals to distance divided by the speed of the lender now this distance is 0 .16 light here so 0 .16 light here so 0 .16 light that was measured by the aliens divided by the speed and speed would be 0 .8c plus 0 .6c now we're going to solve this and we'll get our answer as 0 .114 year so this is the answer for the c part now next we'll be solving the d part now in the d part we have to calculate the kinetic energy as measured in the earth's reference frame and the expression for it is given by kinetic energy equals to 1 over under root 1 minus ux square over c square minus 1 times m c square now we have all the values we can plug it here and this m is the mass of the lender which we are given as m is equals to 4 times 10 to the power 5 kg this is given in the equation now you can put the values so 1 over under root 1 minus ux squared that we have calculated earlier as 0 .946 c whole square divide by c square minus 1 times m we have 4 times 10 to the power 5 and the value of c we have three times 10 to the power 8 meter per second this is a speed of light now we're going to simplify this and we'll get our answer as 7 .5 times 10 to the power 22 june so this is the required kinetic energy that was asked in the equation...

Please give Ace some feedback