A particle moves according to the equation x=10 t^2 where x...

A particle moves according to the equation $x=10 t^{2}$ where $x$ is in meters and $t$ is in seconds. (a) Find the average velocity for the time interval from $2.00 \mathrm{s}$ to $3.00 \mathrm{s}$. (b) Find the average velocity for the time interval from 2.00 to $2.10 \mathrm{s}$.

Key Concepts

Position Function

This is a mathematical expression that defines the location of an object as a function of time. It serves as the basis for analyzing the motion of the object and allows one to derive other properties such as displacement, velocity, and acceleration.

Displacement

Displacement represents the change in the position of an object over a given time interval. It is calculated as the difference between the final and initial positions and is a key component in determining average velocity.

Average Velocity

Average velocity is defined as the total displacement divided by the total time interval. It provides a measure of the overall rate of change of position over a specified period, regardless of the details of the intermediate motion.

Time Interval

A time interval specifies the period over which a measurement, such as average velocity, is calculated. It is essential to correctly determine the start and end times of the interval in order to accurately compute the displacement and the corresponding average velocity.

Transcript

00:01 Today we're talking about motion and want to mention again, uh, in our problem, we are asked to find the average velocity over a certain time interval, uh, and to find the average velocity is given by this equation right here, which says the velocity is equal to the distance divided by the time and were given the time intervals in the question.

00:25 And we need to find the distance, uh, travel during this time intervals and it's given by this top equation x is equal to 10 times t squared, so the first time interval is x at two seconds.

00:41 So that's 10 times to square is equal to 40 meters and then x at three seconds, which is equal to 10 times three squared, which is equal to 90 meters.

01:00 So the total distance traveled over this time interval is 90 meters minus 40 meters.

01:08 So that's 15 meters traveled and the time interval is three seconds minus two seconds.

01:15 The difference between those which is just one second.

01:18 So the average velocity over the time interval, uh, 2 to 3 seconds is 50 meters per second.

01:31 That's our first answer.

01:34 Now we're asked to find the average velocity over different time.

01:38 Interval.

01:40 Let's go back a little bit.

01:48 So it wants to know the average velocity over the time interval.

01:53 Two seconds to 2.1 seconds.

01:55 Well, we already know the distance at two seconds...