Let's examine whether or not the law of conservation of moment...

Question

Let's examine whether or not the law of conservation of momentum is true in all reference frames if we use the Newtonian definition of momentum: p_x = mu_x. Consider an object A of mass 3m at rest in reference frame S. Object A explodes into two pieces: object B, of mass m, that is shot to the left at a speed of c/2 and object C, of mass 2m, that, to conserve momentum, is shot to the right at a speed of c/4. Suppose this explosion is observed in reference frame S' that is moving to the right at half the speed of light. Use the Lorentz velocity transformation to find the velocities and the Newtonian momenta of B and C in S'.

Accepted Answer

Hey, everyone in this problem, we're told that in an inertial reference frame, S A spacecraft X of mass 10 M is stationary without any external influence, the spacecraft undergoes a spontaneous vision event splitting into two fragments. So the first fragment spacecraft Y with mass two M is propelled to the right with a velocity of 0.6 C relative to S while spacecraft Z with mass eight M is driven to the left with a velocity of 0.15 C relative to S A in order to ensure the conservation of momentum. Now we have an observer situated in another inertial frame S prime which is moving rightward with a speed of 0.3 C relative to the original reference frame. S. So we're told to use the Laurent velocity transformation and determine the velocities and the Newtonian momentum of Y and Z in reference frame S prime, we're given four answer choices here. Each of them containing different values for these velocities. VYVZ and the momenta py and PZ in frame S prime. We're gonna come back to these answer choices as we work through this problem. So to start off, hey, we were given a lot of information. So let's write out everything we know. So in reference frame s we know that the velocity V of spacecraft Y so Vy is 0.6 C, OK. We're gonna take to the right to be positive. So this is going to the right. So this is going to be a positive velocity 0.6 C. Now for spacecraft ZVZ, the velocity is going to be 0.15 C but because it's going to the left, it's going to be negative the VZ negative 0.15 C. OK. So those are the two velocities in frame S and then in frame S prime, we wanna figure out what the corresponding velocities are. So what is uh Vy prime? And what is VZ prime? Now, the other piece of information we're told in the speed at which frame of reference S prime moves relative to the frame of reference S OK. So we know that the speed of S prime relative to s it's going to be this V value of 0.3 C. All right. So that's everything we've been told in this problem. We're gonna use the Laurent velocity transformation like we've been told and recall that the equation there is going to be U prime is equal to U minus V divided by one minus UV, divided by C squared. Now, in this equation, the vs are RBK the speed of S prime relative to U prime is going to be the speed of the object in frame S prime U is going to be the speed of the object in frame S and then we have C which is our speed of light, that constant value that we know. Hey, so this is our general equation and we're gonna work it out for Vy. We're gonna work it out for VZ. So let's start with the Y Yeah, we're gonna do this in green. So for Vy, and what do we have? Well, Vy prime, according to our Lawrence transformation is going to be equal to Vy minus V divided by one minus Vy multiplied by V divided by C squared. We know all of these values. So let's go ahead and substitute them in and we get that Vy prime, it's going to be equal to 0.60 minus 0.3 C divided by one minus 0.6 C multiplied by 0.3 C divided by C squared. Now, we know the value of C, that's our speed of light. We could substitute that in. But notice that we don't need to, here, we have ac multiplied by C that gives us C squared, that will divide out with our C squared in that denominator. Um And so we don't even need to substitute that value in, at this point. We're gonna get Vy prime when we simplify is equal to 0.3 C divided by one minus 0.18. And if we work this out on our calculator, we get a VY prime value of about 0.365 854 C. OK. So this is gonna be the speed of fragment or the velocity of fragment Y spacecraft Y in reference frame S prime. OK. So this is one part of the problem. Now, along with the velocity, we also want to find the moment. OK. So let's go ahead and do that right now. While we're dealing with this, why value and recall that the momenta py prime is going to be equal to the mass. So in this case, my, the mass of Y multiplied by the velocity. And in this case, the velocity we want is the one we just calculated VY prime. You know that mouse is the same in this frame of reference. And so our mass is going to be two M and multiplied by that velocity is 0.365854 C. And when we work this out, we get a momentum of about 0.73 1708 multiplied by M multiplied by C. OK. So now we have our velocity and our momentum for spacecraft Y in frame of reference s prime. OK. So we're done half of this question. Let's take a look at our answer choices. Yeah, for the velocity by option A and B have this as 0.4 C which one, we round is what we found. Option C and D have 0.5 C, which is not what we found. So we can eliminate answer choice C and D. Now for our momentum py option A and B have that momentum as 0.7 multiplied by M multiplied by C which again, when we round is what we found. OK. So either A or B could be correct at this point, we're gonna move on to the values for spacecraft Z and see what we find. So for Z, it's going to be the exact same process. OK? We've already done it, we know how to do it. We're just going to use different values. So let's go ahead and do that. We have our Loreen transformation. VZ prime is going to be equal to VZ minus V all divided by one minus VZ multiplied by V divided by C squared. Substituting in our values. VZ prime is equal to negative 0.15 C A. Don't miss that negative. That does change the value. OK. So always pay attention to your signs, always pay attention to the direction so that you know what signs your velocity should have. OK. So negative 0.15 C minus 0.3 C all divided by one minus negative 0.15 C multiplied by 0.3 C divided by C squared. Again, the C squared is going to divide out, we can simplify the rest, we get that VZ squared or sorry, VZ prime is going to be equal to negative 0.45 C divided by one plus 0.45. All right. And we missed the zero there. So let's fix this. So it's not 0.45 it's zero point 045. There we go. And when we work this out on our calculator, we get that VZ prime is going to be equal to negative 0.430622 C. OK. So now we have the velocity of spacecraft Z in frame of reference S prime, we're left with one thing left to do and that is the momentum of spacecraft Z in this frame of reference. And so PZ prime going to be equal to the mass MZ multiplied by the velocity VZ prime. This is equal to eight M multiplied by negative 0.430622 C. OK. Again, don't forget that negative we're dealing with the momentum. So this is a vector. So the sign and direction matters and when we work this out negative 3.44 five multiplied by M multiplied by C. All right. So we've got everything we need to know all four of those values we were looking for. Let's take a look at our answer choices. We had narrowed it down to A and B because they had that velocity for spacecraft Y of 0.4 C and the momentum for spacecraft Y of 0.7 multiplied by M multiplied by C. Now we want when we round the velocity of spacecraft Z to be about negative 0.4 C and the momentum to be negative three multiplied by M multiplied by C. And so this is going to correspond with answer choice. B thanks everyone for watching. I hope this video helped see you in the next one.

Key Concepts

Conservation of Momentum

Lorentz Velocity Transformation

Newtonian Definition of Momentum

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