The position of a particle moving along the x axis varies in...

The position of a particle moving along the $x$ axis varies in time according to the expression $x=3 t^{2},$ where $x$ is in meters and $t$ is in seconds. Evaluate its position (a) at $t=3.00 \mathrm{s}$ and (b) at $3.00 \mathrm{s}+\Delta t$. (c) Evaluate the limit of $\Delta x / \Delta t$ as $\Delta t$ approaches zero, to find the velocity at $t=3.00 \mathrm{s}$.

Key Concepts

Position Function

A position function describes the location of an object at any given time. In kinematics, it is used to model the motion of an object along a coordinate axis and is essential for determining other motion parameters such as velocity and acceleration.

Finite Difference Method

This method involves calculating the change in position over a finite time interval (?x/?t). It provides an average rate of change over that interval and serves as an approximation to the instantaneous rate of change when the time interval is very small.

Instantaneous Velocity

Instantaneous velocity is the rate of change of position at a specific moment in time. It is obtained mathematically by taking the derivative of the position function with respect to time, which provides a precise description of the object's speed and direction at any instant.

Limits and Derivatives

The concept of a limit is used to rigorously define the derivative of a function. In the context of kinematics, taking the limit of the finite difference of position (?x/?t) as the time interval approaches zero yields the instantaneous velocity, linking calculus directly to the analysis of motion.

Transcript

00:01 Hello everyone.

00:01 Today we are finding a particle's position, and that position is given by the function, x, excuse me, x is equal to 3t squared, where x is in meters and t is in seconds.

00:17 So the first problem is, where is the particle? when t is at 3 seconds? well, we just plug it in, and we get 3 times 3 squared, which is equal to 27 meters.

00:30 Excuse me 27 the next the next part of the problem is to figure out the position of the particle when t is equal to 3 plus delta t well we just plug this in again so we have three times 3 plus delta t squared which is equal to 3 times 3 plus delta t squared which is times 3 plus delta t and if we foil it out we foil these two terms out we would get let's see 9 plus 3 delta t plus 3 delta t plus delta t squared all times 3 we can combine the these two to get 6 delta t.

01:48 So we have 3 times 9, which is 27, plus 3 times 6 delta t would be 18 delta t, plus 3 delta t squared.

02:03 Equals x whenever t is equal to 3 plus delta t.

02:08 That's our answer for the second part of the problem.

02:17 And the last part of the problem just asks us as the delta t pro to zero find take the limit as delta t pro to zero of delta x over delta t all this really means is to take the derivative of this function and in order to do that where it's the simple power rule so we would have three times two times t two minus one and in case you're wondering what i just did there the derivative the power rule is the variable, in this case is t.

02:59 It's exponent.

03:00 We bring it down and multiply it...

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