For the particle described in Exercise 7 , sketch a graph of...

Key Concepts

Relationship Between Acceleration and Velocity

The particle's speed behavior is determined by the signs of its velocity and acceleration. When the velocity and acceleration have the same sign, the particle is speeding up; when they have opposite signs, the particle is slowing down. Analyzing their signs in specific time intervals on the graph provides insight into the particle’s dynamic behavior.

Acceleration Function

The acceleration function represents the rate of change of the velocity function over time. It is obtained by differentiating the velocity function with respect to time and can be visualized through its graph. Understanding its shape, sign, and behavior over different intervals helps in analyzing the motion of a particle and determining when acceleration is positive, negative, or zero.

Constant Speed Condition

When a particle is traveling at a constant speed, its acceleration is zero. This indicates that there is no change in the velocity over time, meaning that the velocity function is constant, which is reflected on the graph as segments where the acceleration function touches or remains at zero.

Transcript

00:01 Okay, so for this problem, we are asked to find, so we're asked to sketch the graph of a of t, and we're asked where is the position, where is the particle speeding up, where is the particle slowing down, and when is it traveling at a constant speed, and we need to deduce all of this information given v of t, so the velocity of this particle.

00:24 So recall that a of t, so our acceleration is just equal to the v -pri -t.

00:31 Of t.

00:32 So the slope of r v.

00:34 T.

00:36 So what does, where is this particle speeding up mean? that means where is it concave up or where is, so it would mean where is it concave up if we were given the p of t where we could just see if it's concave up or concave down.

00:52 So the original function.

00:53 But since we're given v of t, we want to know where is v of t positive.

00:59 Where is the slope positive? where's b prime of t, greater than zero, essentially? and so there's only one place that the slope is greater than zero.

01:11 And so that's on zero to one.

01:14 So as we're going up, this is the only positive.

01:18 This one is a negative, and this one is also negative because we're going down.

01:23 So that does answer our next question, which would be, where is this slowing down? or where is the slope negative.

01:34 And that would be from 3 .5, union, 7, 8.

01:40 And when is it traveling at a constant speed? this is actually traveling at a constant speed the entire time.

01:48 Because this, our slope up here is at a constant rate.

01:52 Our slope here is constant.

01:54 Slope here is constant.

01:56 This is constant and this is constant.

01:59 It's all linear.

02:00 So that means everywhere.

02:08 And so now if we were to graph this function, we would want, so since it's all constant, it's all going to be step functions essentially...

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