A new car is tested on a 200-m-diameter track. If the car speeds up at a steady 1.5 m/s^2, how long after starting is the magnitude of its centripetal acceleration equal to the tangential acceleration?

Video transcript

Welcome back everybody. We are making observations about the speed of a particle. And we are told a couple of different things here. We are told that it initially starts off from rest here and we are told that it is rotating in a circle with a radius of .5 m. Now it has a tangential acceleration of 0.25 m per second squared. And we are tasked with finding the time at which the centripetal acceleration is equal to the tangential acceleration. So looking at all these values, I kind of want to relate all these values in the same formula. And a kid a Matic formula comes to mind here that says that our final angular velocity is equal to our angular acceleration times time plus our initial angular velocity actually going to subtract this angular velocity from both sides and divide both sides by our angular acceleration. You'll see that all this cancels out and we get that our time is going to be equal to our final angular velocity minus our initial angular velocity all divided by our angular acceleration. Now, we can use this property to find these two remaining values. So let's go ahead and do that. Starting off with the easiest here is going to be angular acceleration. This is just our tangential acceleration divided by our radius simply .25, divided by .5 giving us an angular acceleration of .5 radiance per second squared. Remember we start out from rest, we now have our angular acceleration. We just need to find what our final angular velocity is going to be here. Well, we know that centripetal acceleration is simply equal to omega squared times our radius. And this is going to be our final angular velocity that we are looking for. I'm gonna divide both sides by our radius. So that's going to get rid of this term right here. And then I will take the square root of both sides getting rid of this power right here. So now you can see we have this formula. So let's go ahead and plug in some values and here's why we can plug in some values. I'm going to substitute the centripetal acceleration with our tangential acceleration because of this property right here. So let's go ahead and do that. We have that. Our final angular velocity is equal to our tangential acceleration of 0.25 divided by our radius of .5 giving us .71 radiance per second. Great. So now we have our final angular velocity, our initial angular velocity and our angular acceleration. So now we can go ahead and use this formula right here to find the time we have that our time is equal to 0.71 minus zero, divided by 00.5, giving us a final answer of 1.45 seconds corresponding to our answer. Choice of D, My apologies here. Just 1.4 seconds, just 1.4 seconds. Responding, answer choice. D thank you all so much for watching. Hope this video helped. We will see you all in the next one.