A race car starts from rest and travels east along a straight and level track. For the first 5.0 s of the car's motion, the eastward component of the car's velocity is given by vx(t) = (0.860 m/s^3)t^2. What is the acceleration of the car when vx = 12.0 m/s?

Video transcript

Welcome back everybody. We are making observations about a truck and we are told that the westward component of its velocity is given by this equation right here. And we are tasked with finding what is the acceleration of the truck when the velocity of the truck is negative 17m/s. Well, here's what we're going to have to do since we are given velocity as a function of time. We know that the derivative of this is equal to our acceleration as a function of time. So we need to find this equation right here, but we are going to need to know what time to plug in here and luckily we're given a velocity and were given a velocity as a function of time so we can use those two in tandem to solve for our tea. So let's go ahead and first find this formula right here and then we'll move on to our t. So the derivative of our velocity as a function of time is equal to our acceleration. So let's take the derivative right here we get that. This too is going to come down. So this will be two times negative 2.134 times T giving us negative 4.26, 84 hour acceleration as a function of time. Now, what time will we be plugging in here? Well, here's what we're gonna do. We're given that our velocity of interest is negative 17 m per second. And we are also told that this is just also equally represented by this function of negative 2.134 T squared. So here's what we are going to do. Let's first divide both sides of this equation by negative 2.134. This gives us that T. Is equal to 17/ 0.134. And my apologies, T squared Is equal to that. So then I'm going to take the square root of both sides on the left here, this radical and this power is going to disappear. And we get that our time of desire is equal to the square root of that. Which when you plug into our calculator we get 2.82 seconds. Great. So now that we have found our time and our acceleration as a function of time, let's plug this into our equation to find our desired acceleration. So our acceleration at a time of 2.82 seconds is going to be equal to negative 4.268 times 2.82. Giving us a final answer of negative 12 point oh four m per second squared corresponding to our answer choice of C. Thank you all so much for watching. Hope this video helped. We will see you all in the next one