Suppose an object moves along a line at 15 m/s, for 0 ≤ t

Question

Suppose an object moves along a line at 15 m/s, for 0 ≤ t < 2 and at 25 m/s, for 2 ≤ t ≤ 5, where t is measured in seconds. Sketch the graph of the velocity function and find the displacement of the object for 0 ≤ t ≤ 5.

Accepted Answer

Welcome back, everyone. In this problem, a car travels at a constant velocity of 12 m per second for a T greater than or equal to 0 but less than 3 seconds, and then at 20 m per second for a T greater than or equal to 3 but less than or equal to 7 seconds. Graph the velocity function and determine the total displacement of the car from T equals 0 to T equals 7 seconds. Now let's focus on the first part of this problem. Let's first graph the velocity function of our cars travels. Now what do we already know? Well, we're told first it travels at a constant velocity of 12 m per second for a t greater than or equal to 0 but less than 3 seconds. So here you notice that this is the first of two parts that our problem has. Now we're talking specifically about our first part, but the velocity function is going to be a piecewise function. So if we were to graph this first part, it goes from T equals 0 to T equals 3, and it's at 12 m per second. So we can draw our line from T equals 0 to T equals 3, and we know that it includes 0, so we will use a solid point at 0, 12, OK. But It does not include 3, so we will use a hollow point at 3:12. Next, we're told, I, you know what, I should have shaded that in blue. OK, it would make more sense to do that. And now in the second part of our problem, we're told it's traveling at 20 m per second for T greater than or equal to 3 but less than or equal to 7 seconds. So now if we go back to our graph, it includes 3 and 7, so we go up to 20 m per second and at 320, we'll have a solid point. And then at 7:20 we'll also have a solid point. So here is the graph of our velocity function. Next, let's see if we can determine the total displacement. What do we know about displacement in a velocity time graph? Well, the displacement is the area under the velocity time graph between T equals 0 and T equals 7. So if I were to use a line here, let me use this line in black, OK. This dotted line Then here, OK. You'll probably know. That underneath this will be our shaded region. So if I put another line here where T equals 3, OK, because now you'll notice that this shape is made up of two rectangles, then this area shaded in blue is going to be the displacement from T equals 0 to T equals 3. And then this area shaded in red is going to be the displacement between T equals 3 and T equals 7. OK? So let me just shade that here. So this is going to be our displacement. Now that we have that, then we can find the areas of both our rectangles to find the total displacement. Now we know that the area of a rectangle is equal to its height multiplied by its base, so for the displacement. From T equals 0 seconds to T equals 3 seconds, OK. Then it's going to be equal to the velocity multiplied by the change in time, given it's a constant velocity. So that's 12 m per second, multiplied by 3 minus 0 seconds. That's 12 multiplied by 3, which is going to be equal to 36 m. Next for our displacement for the area shaded in red, that is the displacement from T equals 3 to T equals 7 seconds. It's going to be equal to V2 multiplied by the change in T2. Now we're just using this to refer to our second velocity and time. So that is going to be 20 m per second, multiplied, and I just realized I should have also put my multiplication sign in the first case. So that's 20 m per second, multiplied by 7 minus 3 seconds. That's 20 multiplied by 4, which equals 80 m. Thus, this tells us. That the total displacement. Is going to be equal to the sum of those displacements which is 80 m. 80 m, yes, sorry, plus 36 m. Which is equal to 116 m. Thus, this is the total displacement of the car from T equals 0 to T equals 7 seconds. Thanks a lot for watching everyone. I hope this video helped.

Suppose an object moves along a line at 15 m/s, for 0 ≤ t < 2 and at 25 m/s, for 2 ≤ t ≤ 5, where t is measured in seconds. Sketch the graph of the velocity function and find the displacement of the object for 0 ≤ t ≤ 5.

Key Concepts

Velocity Function

Displacement

Graphing Piecewise Functions

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