A cat walks in a straight line, which we shall call the x-axis, with the positive direction to the right. As an observant physicist, you make measurements of this cat's motion and construct a graph of the feline's velocity as a function of time (Fig. E2.30). (c) What distance does the cat move during the first 4.5 s? From t = 0 to t = 7.5 s?

Video transcript

Hi, everyone. Let's get started. So you take a jog in the morning as part of your routine exercise. A colleague uses a coordinate system where the motion is parallel to the X axis. And to the right is the positive X direction. The colleague makes measurements of your motion and constructs a graph of velocity as a function of time, determine the distance you covered in the first three seconds and from two seconds to six seconds. So uh given the graph of velocity, we know the area under the curve is going to be our distance traveled. So I'm going to draw some shapes on our graph here that we can figure out the area of and the added area of those shapes together should give us the distance traveled or the area under the curve. So we have a rectangle and a triangle here. So the area of our rectangle is of course going to be base time site. So the triangle or the rectangle extends from zero here to negative one. So our base is going to be negative 1.0 seconds. And we're gonna multiply that by the height which is going to be positive three. Um This is actually meters per second and this is three seconds. So that's going to give us the area of our rectangle as 3.0 m. And then we also have a, a small uh triangle here, a little right triangle. And uh we can take uh one half base times height to get the area of that. So we can see it starts at negative four stops at negative one. So we have a base of negative three and then the height of the triangle extends from 0 to 3. So we'll multiply that by positive three, that will give us a total of negative 4.5 m. And since we need to add these together to get the total area for the 3.0 seconds period, we get, we get negative 7.5 m and wanna take the absolute value of that because our distance is positive here. So we actually just have a distance traveled to 7.5 m. Now, uh let's uh go ahead and analyze our answers real quick to see if we can eliminate any. So because in the first three seconds, we know we traveled 7.5 m, we can cross out anything that doesn't have that. So let's see. A does have 7.5 m for the first three seconds. So that would be correct. B does not C does not and D does not actually. So we can conclude that those must not be correct. And by process of elimination, answer a time for the distance traveled from the first three seconds is 7.5 m and the distance traveled from two seconds to six seconds is 4.0 m. Thank you very much.