Question

Romeo sets off in his spaceship to marry Juliet, who lives on the planet Lovelon, 10 light-years away from earth. He can accelerate at 15 m/s2, and must decelerate at the same rate, in order to stop at Lovelon. Unknown to him, Juliet has taken an earlier shuttle back to Earth and she will wait there. Since there is no communication, Romeo will have to travel back again! Romeo left yesterday, on April 30, 1997. What is the earliest year in which they can be wed? Romeo and Juliet are both classical persons, and they ignore any relativistic effects! 1 light-year is about 9.46 x 10^15m.

Answer 1

please help, i'm stuck on this question

Answer 2

So the fastest Romeo can get to Lovelon is by accelerating half the distance and then decelerating half the distance. The deceleration will be equal in magnitude as the acceleration (but opposite in sign of course). Romeo will have to do the the round trip which means covering the half-distance to Lovelon 4 times. The equation to start with is:\[x=x _{0}+v _{0} t+\frac{ 1 }{ 2 }at ^{2}\]where x is the total distance traveled; x0 is the starting distance; v0 is the starting velocity; a is the acceleration; and t is the time. Now we know that x0=0 (i.e. he hasn't traveled any distance at the start of his trip) and v0=0 (he's starting from a dead stop). So we're left with:\[x _{0.25}=\frac{ 1 }{ 2 }at _{0.25}^{2}\]If the time of the entire trip is t, the time of accelerating to Lovelon is t0.25, and that's one quarter of the trip. That's what t0.25 signifies. Like wise x0.25 indicates ¼ of the total distance traveled. Now we can simplify that equation:\[t _{0.25}=\sqrt{\frac{ 2x _{0.25} }{ a }}\]Now total time is given by:\[t=4*t _{0.25}\]so the final equation is:\[t=4\sqrt{\frac{ 2x _{0.25} }{ a }}\]Again:\[x _{0.25}=\frac{ x }{ 4 }=\frac{ 9.46*10^{15}m }{ 4 }\].

It's straightforward, now, to find the answer. Note that you'll have to change your answer from seconds to years.

Answer 3

how do we know that the fastest way is to accelerate for half the distance and deccelerate for the other half?

Answer 4

ok, i get it now, but i really dont get why he accelerates for half the distance and deccelerate for the other half.

Answer 5

Well, if he can only accelerate at 15m/s, that means using maximum power, he can only decelerate at that same rate. In fact it says in the problem that he decelerates at the same rate that he accelerates.

Answer 6

I can tell you that Juliet is going to look a lot different when they finally meet again.

Answer 7

Actually, let me correct one error. x is given by this:\[x=\frac{ x _{0.25} }{ 2 }=\frac{ 9.46*10^{15}m }{ 2 }=4.73*10^{15}m\]x0.25 is one quarter of the trip or one half of the distance from Earth to Lovelon. So, given my error, I should probably let you know the trip is going to take about 76.4 years.

If you're curious, you should know that the max speed achieved is 1.78 times the speed of light, which of course is not possible.