Give the positions s=f(t) of a body moving on a coordinate...

Give the positions $s=f(t)$ of a body moving on a coordinate line, with $s$ in meters and $t$ in seconds. a. Find the body's displacement and average velocity for the given time interval. b. Find the body's speed and acceleration at the endpoints of the interval. c. When, if ever, during the interval does the body change direction? $$s=-t^{3}+3 t^{2}-3 t, \quad 0 \leq t \leq 3$$

Give the positions $s=f(t)$ of a body moving on a coordinate line, with $s$ in meters and $t$ in seconds.
a. Find the body's displacement and average velocity for the given time interval.
b. Find the body's speed and acceleration at the endpoints of the interval.
c. When, if ever, during the interval does the body change direction?
$$s=-t^{3}+3 t^{2}-3 t, \quad 0 \leq t \leq 3$$

Key Concepts

Displacement

Displacement is the net change in position of an object over a given time interval, calculated as the difference between the final and initial positions. It is a vector quantity, meaning it has both magnitude and direction, and it helps determine how far and in which direction the object has moved from its starting point.

Average Velocity

Average velocity is defined as the displacement divided by the time elapsed during the interval. It provides a measure of the overall rate of change of the position of the object over the time period, incorporating both the magnitude and the direction of movement.

Instantaneous Velocity

Instantaneous velocity is the derivative of the position function with respect to time. It gives the rate at which an object's position is changing at any specific moment in time. This concept is essential for understanding how the object's speed and direction evolve continuously.

Speed

Speed is the magnitude of the instantaneous velocity and is always non-negative. It represents how fast an object is moving regardless of its direction, serving as a scalar measure of motion.

Acceleration

Acceleration is the derivative of the velocity function with respect to time, representing the rate at which velocity changes. It is fundamental in analyzing how quickly an object speeds up, slows down, or changes direction.

Change of Direction

A change of direction occurs when the instantaneous velocity changes its sign, indicating a reversal in the motion's direction. Identifying such points typically involves finding the zeros of the velocity function and analyzing the sign changes around these critical points.

Transcript

00:01 For this problem here, we are first given this equation, which equals our position, and on the interval from square three seconds.

00:16 So for part a, we're first being asked, what is our displacement? in this case, delta s and what is our average velocity? so to find our displacement, this would be a final versus minus s initial.

00:33 So the final position minus the initial position.

00:37 To find that, we just plug in our endpoints.

00:40 So for s final, this would be negative 27 plus 27 minus 9.

00:47 So this would basically be negative 9.

00:49 For s initial, because we have zero here, and all these values have a t in them, it'll just be 0.

00:55 So negative 9 minus 0, which is negative 9.

00:58 This is our displacement.

01:01 And for our average, velocity.

01:04 This is, to get this, we just plug in delta s over delta t.

01:10 This is what average velocity is the change in displacement over change of time.

01:15 So for this problem, this for us, it would be negative of mine over 3, which gives us negative 3 meters per second, and for this one meters.

01:28 So for part b now, moving into part b, we're being asked at the endpoints 0 and 3.

01:36 What's the speed and what is our acceleration? so to find our speed, it's essentially the absolute value of velocity.

01:46 And to find our velocity, we can look at the position function given to us, and we can take the derivative of it a bit, because the derivative of position is velocity.

01:58 So this would give us negative 3t squared plus 6t minus 3, right? negative 3 t squared.

02:06 Plus 6 t minus 3.

02:09 And plugging in 0 and 3, for 0, we get that this is 3 meters per second, because these have the t's.

02:19 And then we're left with a negative 3, but we need to take the absolute values, let's give us 3.

02:23 And for 3, plug in the values, we get negative 27, plus 18 minus 3.

02:29 This gives us negative 12, right? but then we need to make an absolute value for this.

02:35 So this gives us 12 meters...

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