A proton has momentum with magnitude p0 when its speed is 0.400c. In terms of p0, what is the magnitude of the proton's momentum when its speed is doubled to 0.800c?

Video transcript

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. A neutron moves at 0.75 C. Its momentum is P if the speed is divided by 1.5 express the new momentum in terms of P Awesome. So we're given some multiple choice answers. Let's read those off to see what our final answer might be. A is 1.15 PB is 0.509 PC is 1.6 P D is 0.471 P and E is 0.667 P. Awesome. So first off, we need to recall what the relativistic momentum formula is. We need to use this formula because the neutron speed is a substantial fraction of the speed of light. So let's call the relativistic momentum formula equation one. And it states that the momentum P equals the relativistic factor. It's also called Lourens factor denoted as gamma multiplied by the mass of the neutron M multiplied by V which represents the speed of the neutron. We also need to remem recall the equation for the Lourens factor or the relativistic factor. And that states that one divided by the square root of one minus V squared divided by C squared Awesome where C is the speed of light in this case? Awesome. So we can use the relativistic momentum formula to determine what the new momentum will be. So let us write the new momentum as a, as a ratio. And let's call it equation two. So let's do that. So let's denote the new momentum as P subscript new. So the new momentum divided by P equals gamma subscript, new gamma, new multiplied by V new V subscript, new gamma subscript new. So gamma new multiplied by V new sorry gamma, new multiplied by V new, divided by gamma multiplied by V. So to get the new momentum by itself, we need to multiply both sides by P and that will give us equation three. So the new momentum equals P multiplied by V new, divided by V multiplied by gamma new, divided by gamma. Also, at this stage, we can assume that we knew divided by V equals one, divided by 1.5 since it states that speed is divided by 1.5 in the problem. Awesome. So now we can solve for the relativistic factor. So let's do that. And this is not the new relativistic factor. This is just the relativistic factor from the ratio So we're solving for this relativistic factor, we'll do the relativistic factor for the new value in just a second. And we'll call R S our assumption here. Equation four. So equation five, solving for the relativistic factor gamma equals one divided by one minus. And this, the speed V is given to us in the prompt. So the neutron moves at 0.75 C as stated in the practice problem divided by C square. So we need to plug in our known variables which we don't have to plug in the speed of light because as you notice the CS cancel out. So when you plug this into a calculator, noting that the CS cancel out, we get that gamma equals like after we type everything into a calculator, it will be 1.5119. Awesome. So now let's solve, we'll call it the equation six for the new relativistic factor value doing the same thing as above same process here. One divided by the square root of one minus that's slightly different here. But yes, we have to use the new relativistic factor is we have to take into account the speed divided by 1.5. So let's do that. 0.75 C divided by 1.5. And this is all squared. So 0.75 C divided by 1.5 all to the power two divided by C squared. So all like above the, the speed of light, the seas cancel out and we determine that the new relativistic factor when you plug into a calculator, it will give us 1.1547. Awesome. So to solve for the value of the new momentum using equation three, so we can solve for the new momentum using equation three. So let's plug in our known variables using equation three here. So let's recall and use that from above. So the new momentum equals P multiplied by one divided by 1.5. When we plug in our known variables as we did, we determined that V new, divided by V was equal to one divided by 1.5 multiplied by the new relativistic factor, which was 1.1547 divided by just the relativistic factor. The original one, the starting one which was 1.5119. So when you plug that into a calculator, you should get 0.509 P, which is our new momentum. Awesome. We did it. That was a lot of work, but we did it. OK. So that means our final answer is B 0.509 P. Thank you so much for watching. Hopefully, that helped and I can't wait to see you in the next video. Thank you. Bye.