Displacement from a velocity graph Consider t...

Question

Displacement from a velocity graph Consider the velocity function for an object moving along a line (see figure).

(d) Assuming the velocity remains 10 m/s, for t ≥ 5, find the function that gives the displacement between t = 0 and any time t ≥ 5.

Graph showing velocity in m/s over time in seconds, with a constant velocity of 10 m/s from t=5 seconds onward.

Accepted Answer

Hello. In this video, we are given that the velocity time graph of a car moving along a straight road for any time T is on the interval from 0 to 10. We want to find the displacement function for T is greater than or equal to 8, if the velocity remains constant for T greater than or equal to 8. So, what's important here is to note that velocity is constant from the time 8 and above. If we look at that interval on the given graph, that interval exists here, where the value of velocity is going to be 20. Now furthermore, for displacement, we need to know the area underneath the rest of the curve, so we can represent the displacement function to be S of T equal to S of 8, plus the integral of velocity, so it's going to be the integral from 8 to T. A velocity with respect to T. Now, in this case here, because velocity is constant and velocity has a value of 20 from 8 and above, we can represent this integran to be 20 DT. So, let's go ahead and start by evaluating the integral. Now, by taking the antiderivative of a constant function, we'll be left with 20 T evaluated from T to 8. By plugging in the values of the bounds, this is going to give us 20 T minus 20 multiplied by 8, and this is going to further simplify to be 20 T minus 160. So this is going to be the value of the integral underneath the curve from 8 to T with respect to velocity. But what about the value for S of 8? Well, SF 8 is going to be the displacement from 0 to 8. So what we need to do in order to figure out this displacement is we need to find the area underneath the curve from 0 to 8. Now, we can do this by using geometric methods. We will split up the curve where the behaviors change. The behaviors of velocity change at X is equal to 3 and X is equal to 6. So, in order to find the area under the curve from 0 to 3, we're going to use the area of a triangle. This triangle has a height of 30. And has a width of 3. And so that area is going to be 1/2 multiplied by 30, multiplied by 3, which is going to give us the value. Of 45. So this area is going to be 45 units. Next, we will take the area between 3 and 6. This is going to be the area of a rectangle with the height of 30, and a width of 4, and that area is going to be length times which, which is going to be 120. Now the area from 6 to 8 is a bit trickier because we are going to have to use the area of a trapezoid. The area of a trapezoid is 1/2 multiplied by the first height plus the second height multiplied by the width. Now, the width in this case is going to be 2 units. The first height has a height of 30, and the second height has a height of 20. So this area is going to be defined as 1/2 multiplied by 30 plus 20. Multiplied by 2, and this is going to simplify to give us a final area of 50. Now that we've identified the areas underneath the three sections of the curve, we will now take the sum of these areas to define S of 8. And so the sum of these three areas is going to be 45 plus 120 + 50. And this is going to simplify it to be 185. Now that we've identified the values of the integral and SF 8, we can take these values and plug them back in for the displacement function. That is going to be S of T is equal to 185 plus 20T minus 160, and this is going to simplify to give us a displacement function of 20T plus 25, and this is going to be the displacement function for the given velocity and time function. And with that being said, the overall solution to this problem is going to be Is going to be B. So, I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

Displacement from a velocity graph Consider the velocity function for an object moving along a line (see figure).
(d) Assuming the velocity remains 10 m/s, for t ≥ 5, find the function that gives the displacement between t = 0 and any time t ≥ 5.

Key Concepts

Velocity and Displacement Relationship

Integration in Calculus

Piecewise Functions

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