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? $$s=t^{2}-3 t+2, \quad 0 \leq t \leq 2$$

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?
$$s=t^{2}-3 t+2, \quad 0 \leq t \leq 2$$

Key Concepts

Average Velocity

Average velocity is calculated by dividing the displacement by the time interval over which the displacement occurs. It provides a measure of how fast the position of the body changes on average and includes information about the direction of motion, making it a vector quantity.

Change in Direction

A change in direction occurs when the velocity of the body changes its sign. Detecting a point in time where the instantaneous velocity equals zero (and changes sign around that point) helps determine if and when the body reverses its direction during its motion.

Acceleration

Acceleration is determined as the derivative of velocity with respect to time, or equivalently, the second derivative of the position function. It describes how the velocity of the body changes over time, indicating whether the body is speeding up or slowing down, and in which direction.

Displacement

Displacement measures the change in position of the body between two points in time. It is computed as the difference between the final position and the initial position, reflecting the overall change in the body’s location, irrespective of the path taken during the motion.

Instantaneous Speed

Instantaneous speed is the magnitude of the instantaneous velocity, which is the derivative of the position function with respect to time. It represents how fast the body is moving at a specific moment without regard to the direction of motion.

Transcript

00:01 For this problem, they're first asking us what the difference in displacement is or what our displacement is.

00:07 My fact, that our displacement is between 0 and 2.

00:11 So we're giving the function for position as equals t squared minus 3t plus 2.

00:18 So to find the difference in position, we'd plug in the endpoints and then subtract final minus initial.

00:25 So if we plug in 2, we'd get 4 minus 6 plus 2.

00:30 And this would equal 0.

00:32 If we plug in 0, we would get, sorry, dash point here, separating them.

00:38 We would get 0 minus 0 plus 2 equals 2.

00:41 So that means our final minus our initial would be negative 2.

00:47 So our change in displacement would be negative 2 meters.

00:52 And as for our average velocity, we calculate this by finding our final plus our initial divide by 2.

01:00 So to find a velocity, we need to take the derivative of this, and this would give us v of t.

01:09 Once we take the derivative equals 2t minus 3.

01:14 So once we do that, we plug in 0 and 2, and we get that the final one, which is if you find t equals 2, it would be 4 minus 3.

01:25 So that would equal 1 plus the initial one, which be 2 times 0, which is 0, minus 3 would be negative 3 over 2, this value equals negative 2 over 2, which equals negative 1 meters per second...

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