Graphs of the position functions of two particles are shown. where t is measured in seconds. Whe

Graphs of the position functions of two particles are shown....

Key Concepts

Position Function

The position function specifies an object's location as a function of time. In problems involving graphs of position functions, the slope of the curve at any point represents the instantaneous velocity of the particle, meaning that an increasing position graph indicates a positive velocity while a decreasing graph indicates a negative velocity.

Velocity Function

The velocity function, which is the first derivative of the position function, tells us how fast and in which direction the particle is moving. Key to understanding the motion is determining the sign of the velocity. A positive velocity indicates motion in one direction, while a negative velocity indicates motion in the opposite direction.

Acceleration Function

The acceleration function is the derivative of the velocity function (or the second derivative of the position function) and shows how the velocity is changing over time. A positive acceleration means that the velocity is increasing, and a negative acceleration means it is decreasing.

Relationship between Velocity and Acceleration

The concepts of speeding up and slowing down depend on the relationship between the velocity and acceleration. A particle is speeding up if its velocity and acceleration have the same sign, indicating that the speed is increasing in the direction of motion. Conversely, if the velocity and acceleration have opposite signs, the particle is slowing down because the acceleration is acting opposite to the direction of motion.

Transcript

00:01 All right, so in this problem, we are given two graphs which depict the position of some sort of particle.

00:07 And what the problem is asking us to do is find the intervals on which this particle is speeding up and when it is slowing down.

00:13 And so the key to doing this is finding the function's velocity and this acceleration, which we are going to do now.

00:20 But first, i feel like it'll be good to talk about what speeding up and slowing down actually mean in the context of this problem.

00:27 So all functions have a velocity and acceleration.

00:32 When we talk about speeding up and slowing down, we need to remember that speed is different than velocity, where speed can't be negative.

00:38 So speed's the absolute value of velocity.

00:41 So when a particle has a positive velocity, so it's moving forward, and has a positive acceleration, which means it's increasing, the velocity is increasing, that means we're speeding up because we are going in the positive direction and our acceleration is making us go more positive.

00:58 It's increasing our velocity.

01:00 Similarly, if we have a negative velocity and our acceleration is negative, that means we're going backward and our acceleration is decreasing the velocity.

01:10 So it's making it more and more negative.

01:12 So even though the velocity is decreasing, our speed is increasing because we're going faster backward.

01:19 And then for these other ones, it's going to be slowing down because we have a negative velocity and a positive acceleration, it means we're going backward, but the acceleration is making the velocity increase, so it's becoming a smaller, negative number.

01:33 So we're moving backwards slower.

01:35 And same with a positive velocity and negative acceleration.

01:38 We're moving forward, but if the acceleration is negative, it's making us move forward slower.

01:43 So what we need to keep in mind we're doing this problem is that if the signs for the acceleration and the velocity match, that means it's speeding up.

01:51 But if the signs for the acceleration and the velocity don't match, that means our particles are going to be so.

01:56 Slowing down.

01:57 So let's hop over to part a, this first problem, and let's start marking the velocity and the acceleration of this function.

02:05 So for the first second here from 0 to 1, we have a positive velocity because this graph is increasing, and then we have a negative velocity for the next two seconds, and then we have a positive velocity after that, because the part is going up for the first second, down for the next two, and then up for that last second.

02:31 As for acceleration, there's a little mnemonic where concave up looks like a cup, concave down looks like a frown.

02:40 So when it looks like it's smiling, when it looks like a cup, the acceleration is positive.

02:45 When it looks like a frown, the acceleration is negative.

02:47 So we can kind of see the frown for the first two seconds here along this path.

02:55 So that means for these first two seconds we have a negative acceleration, and then we can see the smile for the last two seconds, which gives us a positive acceleration.

03:06 So when it comes to speeding up and slowing down, when you look at when these signs match...

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