A turtle crawls along a straight line, which we will call the x-axis with the positive direction to the right. The equation for the turtle's position as a function of time is x(t) = 50.0 cm + (2.00 cm/s)t − (0.0625 cm/s^2)t^2. (a) Find the turtle's initial velocity, initial position, and initial acceleration.

Video transcript

Welcome back everybody. We are making observations about a train, which I'm just going to represent by this box that is moving to the right along a straight railway track and we are given its position as a function of time by this equation right here and we are tasked with finding what the initial position, initial velocity and initial acceleration are of this train. Well, how are we gonna do this? Well, for all of these initial values, we know that the time is just going to be zero, it's at the very start of it all. So what we're going to need to do is plug the time of zero into our position as a function of time. Our velocity as a function of time and our acceleration as a function of time. We already know this first equation because that is what we're given. But what about these two? Well, we know through the relationships and derivatives, we know that the derivative of position as a function of time is velocity as a function of time and the derivative of velocity as a function of time is our acceleration. So first thing to do is to find these two equations here. So let's go ahead and do that. So the derivative of position with respect to time is going to be our velocity which is equal to remember we're taking the derivative of this equation right here and we'll do it term by term this first term right here is a constant. So it'll just be zero. This second term. This t is just going to be Go away. So this will just turn into a constant. In this last term we are going to use the power rule. The two is going to come down in front and this will multiply what's on the inside the .03-4. And then just leave multiplied by T. On the outside. Great. So now we are going to take the derivative again giving us our acceleration as a function of time. So zero just goes away. This constant right here also goes away. So that's just gonna be zero minus the derivative of this term. The T. Is just going to disappear. So this will just be two times 0.3 to four as our acceleration. I'm just gonna scroll down just a little bit here and now we are ready to find our initial position velocity and acceleration. So, let's start with our position here. Right. We are plugging in a time of zero into this equation way up here and this gives us 25 plus 3.15 times zero minus 0.324 times zero squared because of the zeroes. These terms are just going to disappear and we're just left with an initial position of 25 centimeters now, what about its initial velocity? We are just plugging in t of zero into this equation right here. So this is going to be 3.15-2 times 0.0324 times zero. This second term is just going to disappear, leaving us with just 3.15 cm/s. Great. Now finally moving on to acceleration, this is just going to be minus two times 0.324, giving us negative 0. centimeters per second squared. So now we have found the initial position, the initial velocity and the initial acceleration, and this corresponds to our final answer choice of a Thank you all so much for watching. Hope this video helped. We will see you all in the next one.