Calculate the speed of a proton (m = 1.67 x 10^-27 k...

Question

Calculate the speed of a proton (m = 1.67 x 10^-27 kg) whose kinetic energy is exactly half its total energy.

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A muon with a mass of 1.88 multiplied by 10 to the power of negative 28 kg is moving at a relativistic speed, determine the speed of the muon. In the following cases, I, when its kinetic energy is one third of its total energy and I I when its kinetic energy is 1/4 of its rest energy, OK. So it appears for this particular problem we're asked to solve for two separate answers. For part I, we're trying to figure out the speed of the muon when its kinetic energy is one third of its total energy. And for part I I, we're trying to figure out the speed of the muon in the case that its kinetic energy is 1/4 of its rest energy. Awesome. Now that we know that we're ultimately trying to figure out what the speed of the muon is for two different conditions. Let's read off multiple choice sensors to see what our final answer pair might be noting for parts I and I, I, there all of the speed of the muon is V equals sum expression. So I'm gonna read them off in order from I to part I, I Awesome. So A is the square root of five divided by three multiplied by C and three divided by five, multiplied by CB is the square root of five divided by three multiplied by C and three multiplied by CC is five divided by three multiplied by C and three multiplied by C and D is five divided by three multiplied by C and three divided by five multiplied by C. And in case you don't remember or you're wondering what C is C denotes the speed of light. So it's our some numerical value this fraction multiplied by the speed of light. That's what C represents. Awesome moving right along here. So first off, let us start to solve for part I. So let's make a little note here that we're solving for part I first. So we need to recall and use the relation for the kinetic energy. So we need to recall and use the kinetic energy equation. So let us recall that ke the kinetic energy is for this particular case because we're dealing with part I is equal to one third multiplied by the energy because it's one third, it's total energy. So K equals one third multiplied by E. So now we need to recall and use the relativistic kinetic energy formula which states that ke the kinetic energy is equal to gamma minus one multiplied by M multiplied by C squared fantastic. And let us also recall and use the equation for the total energy and the total energy E is equal to gamma multiplied by M multiplied by C squared. So considering this, when we plug in the value for E into our relativistic kinetic energy equation, we can easily write that gamma minus one multiplied by M multiplied by C squared is equal to one divided by three multiplied by gamma multiplied by M multiplied by C squared. Fantastic. So moving right along here, we now need to simplify and solve for gamma. So when we simplify, we will find that gamma minus one is equal to one third multiplied by gamma. So ultimately using a little bit of algebra to get gamma all by itself, I'm gonna skip a few steps. But you'll know you've done your algebra correctly when you get gamma all by itself and gamma is equal to three halves. Fantastic. Moving right along here. So our next step is we need to relate gamma to the velocity using the Lorenz factor equation, which we should be able to recall the Lorenz factor equation as gamma equals one divided by the square root of one minus V squared divided by C squared Fantastic. So let me plug in our known variables. We will find that gamma, in this case is equal to three halves, which is equal to one divided by the square root of one minus V squared divided by C squared. Fantastic. So now what we're trying to do is we're ultimately trying to take the C expression and we're rearranging it to isolate and solve for V all by itself by using some algebra. And to do that, all you need to do is start by squaring both sides. So once you've squared both sides and you've simplified, you will ultimately find that V is equal to the square root of five divided by three, multiplied by C. And that is our final answer. For part I hooray, we did it. So I skipped some steps in order to make sure we have enough time to solve both problems here. But ultimately, all you need to do is to take your expression initially square both sides and simplify down to its most simple form. And we're trying to rearrange the equation to isolate and solve for V all by itself. Awesome. Now, moving right along here, we could start solving for part I I now which is our second answer. So we're trying to figure out the kinetic energy which is 1/4 of its rest energy. So in order to do that, let us recall and set up our relationship for part I I, which states that for this particular case, that the kinetic energy is equal to 1/4 multiplied by M multiplied by C squared fantastic. And now we need to recall and use the relativistic form kinetic energy formula which once again, we should recall that that is ke is equal to gamma minus one multiplied by M multiplied by C square. So we can therefore go ahead and write that gamma minus one multiplied by M multiplied by C squared is all equal to 1/4 multiplied by M multiplied by C squared. Fantastic. So now like before we need to simplify and solve for gamma, so we do, we find that gamma minus one is equal to 1/4. And then ultimately using a little bit of algebra to get gamma all by itself, we will find that gamma is equal to five fourths or five divided by four. Awesome. So now our next step is we need to relate gamma to the velocity. So when we do that, we will find that gamma is equal to one divided by the square root of one minus V squared divided by C squared. So like before when we plug in our value for gamma, we will find that five fourths is equal to one divided by the square root of one minus V squared divided by C squared Fantastic. So like before we have to rearrange this equation to isolate and solve for V all by itself. So you know that you will know if you've done your algebra correctly, if V is equal to three divided by five or 3/5 multiplied by C. And that is our final answer. Hooray. We did it for part I I Awesome. So a quick recap here, the speed of the muon is V equals square root of five divided by three, multiplied by C for part I. And for part I I, the speed of the muon is V equals three divided by five multiplied by C Awesome. So this corresponds with, when we look at our multiple choice answers, multiple choice answer letter A which once again, for I, the speed of the muon is V equals square root of five divided by three multiplied by C and I I is the speed of the muon is V equals three divided by five multiplied by C. Thank you so much for watching. Hopefully, that helped and I can't wait to see you in the next video I

Calculate the speed of a proton (m = 1.67 x 10^-27 kg) whose kinetic energy is exactly half its total energy.

Key Concepts

Kinetic Energy

Total Energy

Rest Energy

Watch next

Calculate the speed of a proton (m = 1.67 x 10-27 kg) whose kinetic energy is exactly half its total energy.

Key Concepts

Kinetic Energy

Total Energy

Rest Energy

Watch next

Calculate the speed of a proton (m = 1.67 x 10^-27 kg) whose kinetic energy is exactly half its total energy.