Scientists design a new particle accelerator in which protons (mass 1.7 X 10^-27 kg) follow a circular trajectory given by r = c cos (kt^2) î + c sin (kt^2) ĵ, where c = 5.0 m and k = 8.0 x 10^4 rad/s^2 are constants and t is the elapsed time. a. What is the radius of the circle?

Video transcript

Hey, everyone today, we're dealing with the problem using position vectors. So we're being told that a private tech company has built one of the most advanced and powerful particle accelerators ever already uses electricity to quote unquote push the charged particles which are electrons along a circular path, making them go faster and faster. The path to follow is given by a specific equation where R is equal to V cosine Katie squared, I plus V sign Katie squared, J where V is equal to four m and K is equal to seven times 10 to the fourth radiance per second squared. These values are constant and T is the time with this, we're being asked to determine the circles of radius. In other words, we're being asked to determine the magnitude, right. We're going to need the magnitude or the length of this position of this vector that we're given, right. So if we're to do this, if we can use our uh magnitude formula, which uses the square root of the sum of the squares of the terms here. So that will mean the magnitude of R, the radius that we're looking for will therefore be the square root of, excuse me, the square root of a Kassian Katie. Oops K or sorry, not a V. It's V Kassian K T squared plus the sign Katie squared. And we square both of these terms because again, we're taking the square root of the sum of squares. So this is give us the square root of oops. Alright. That little leader squirt of, come on a squirt of V squared, cosign square K T squared plus V squared. That's V squared sine squared K T squared, which we can simplify and get oops V squared into Kassian squared, K T squared plus sine square K T squared. So using the identity that Kassian squared T squared plus sine squared K T squared is equal to one, you two are sort of calculus slash Trigana metric identities. We can therefore just say that the magnitude of our will just be equal to the square root of B squared. And we know that V is equal to four. So this will just simply be the square of four squared, Which would give us four US four, which means The radius of this circle. The magnitude of its position vector is therefore four m or answer choice B I hope this helps. And I look forward to seeing you all in the next one.