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

00:02 Here we have the graph of a position function, and we want to know when the particle is speeding up and when it's slowing down.

00:09 It'll be speeding up when the velocity and acceleration have the same sign, either both positive or both negative, and it will be slowing down when they have different signs, one positive and one negative.

00:20 So what we're going to do is break our time intervals down and analyze what's happening with the velocity and acceleration in each interval.

00:28 So we have 0 to 1, 1 to 2, 2 to 3, and 3 to 4.

00:35 Something is changing for each interval.

00:38 So now we're going to find the sign of the velocity and the sign of the acceleration.

00:44 When the position graph is increasing, the velocity is positive.

00:48 So we see it increasing from 0 to 1, and we see it increasing from 3 to 4.

00:54 When the position equation is decreasing, the velocity is negative.

00:58 So we see it decreasing from 1 to 3.

01:02 Now the acceleration is the second derivative of position, and the second derivative relates to the concavity of the curve, concave up versus concave down.

01:12 If the curve is concave down, the second derivative will be negative, and if the curve is concave up, the second derivative will be positive.

01:19 So we see that this curve is concave down from 0 to 2.

01:23 So we have a negative acceleration, and then it's concave up from 2 to 4.

01:27 So we have a positive acceleration.

01:30 Now let's look and see when both of these signs are positive.

01:34 Velocity and acceleration are both positive from 0 to 4, and they're both negative from 1 to 2.

01:39 So that's the indication of speeding up...

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