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? $$s=6 t-t^{2}, \quad 0 \leq t \leq 6$$

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?
$$s=6 t-t^{2}, \quad 0 \leq t \leq 6$$

Key Concepts

Change of Direction

A change of direction in an object's motion occurs when its instantaneous velocity changes sign. This is an important concept in kinematics that indicates a reversal in the direction of motion, typically identified by finding the roots of the velocity function over the interval.

Speed

Speed is the magnitude of velocity, meaning it is a scalar quantity that measures how fast an object is moving regardless of its direction. It is obtained by taking the absolute value of the instantaneous velocity.

Acceleration

Acceleration is the rate at which the velocity of an object changes with time. It can be found by differentiating the velocity function or, equivalently, by taking the second derivative of the position function with respect to time. This concept is crucial for understanding how quickly an object is speeding up or slowing down.

Average Velocity

Average velocity is defined as the total displacement divided by the total time taken. It is a vector quantity, meaning it has both magnitude and direction, and it gives a measure of the overall rate of change of position during the interval.

Instantaneous Velocity

Instantaneous velocity is the derivative of the position function with respect to time. It represents the speed and direction of the object at any specific moment and is key to understanding how the object’s motion is changing at each point in time.

Displacement

Displacement is the overall change in position of an object over an interval of time. It is calculated by subtracting the initial position from the final position. Unlike distance traveled, displacement accounts only for the net change regardless of the actual path taken.

Transcript

00:01 So for this problem, we're given this equation for position right here.

00:04 And for part a, they're asking us, okay, so what is our displacement? what is our change in position or change in s right there? and then to find this, basically we find s final minus s initial, according to our time frame, which is six seconds to zero seconds.

00:22 So find that we can just plug in the value at six for to get s final.

00:27 And this would be 36 minus 36, so it would be first 0.

00:32 And for us initial, plug in 0, we get 0 again.

00:35 So change in position is 0 meters.

00:40 Now for the second part of part a, it's asking us, what is our average velocity? so our average velocity can refine by our change in position over change in time.

00:52 And since we found our change in position right here is 0, we can just divide this by our time.

00:58 This again would give us zero.

01:00 So this would be zero meters per second.

01:05 Now moving on to part b, the question asks us, okay, at our end points, what is the speed? we'll call this s -not.

01:14 And what is our acceleration? so to find our speed, we're going to find basically the opposite value of the velocity.

01:25 And to find our velocity, looking at our position function, we can take the derivative of this to find the function for velocity or equation for it.

01:36 So this becomes 6 minus 2t.

01:41 And at 0, this value is 6 meters per second...

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