The position of a particle along the x axis depends on the...

The position of a particle along the $x$ axis depends on the time according to the equation $x=A t^{2}-B t^{3}$, where $x$ is in meters and $t$ is in seconds. (a) What SI units must $A$ and $B$ have? For the following, let their numerical values in SI units be $3.0$ and $1.0$, respectively. (b) At what time does the particle reach its maximum positive $x$ position? ( $c$ ) What total path-length does the particle cover in the first 4 seconds? ( $d$ ) What is its displacement during the first 4 seconds? ( $e$ ) What is the particle's velocity at the end of each of the first 4 seconds? $(f)$ What is the particle's acceleration at the end of each of the first 4 seconds? $(g)$ What is the average velocity for the time interval $t=2$ to $t=4 \mathrm{~s}$ ?

Key Concepts

Dimensional Analysis

Dimensional analysis is used to determine the units of physical quantities by ensuring that both sides of an equation have the same dimensions. In this context, it helps assign the correct SI units to the constants in the kinematic equation so that the equation describing position as a function of time is dimensionally consistent.

Kinematic Equations

Kinematic equations describe the motion of a particle as a function of time, typically including relationships for position, velocity, and acceleration. These equations are fundamental in physics for predicting how objects move under various conditions and involve taking derivatives to transition between position, velocity, and acceleration.

Critical Points and Optimization

Finding the maximum or minimum positions in a particle's trajectory involves determining critical points by setting the derivative (velocity) to zero. This method, often involving the first derivative test, is essential in analyzing the motion to identify turning points where the position is at an extreme value.

Path Length vs. Displacement

Path length refers to the total distance traveled by a particle along its trajectory, whereas displacement is the straight-line difference between the initial and final positions. Understanding the distinction between these two concepts is crucial when analysing motions that include changes in direction.

Instantaneous Velocity

Instantaneous velocity is the rate of change of position at a specific moment in time and is determined by differentiating the position function with respect to time. It provides insight into how fast the particle is moving and in which direction at any given moment.

Instantaneous Acceleration

Instantaneous acceleration is the rate of change of velocity with respect to time, found by taking the second derivative of the position function (or the first derivative of the velocity function). It indicates how quickly the velocity of the particle is changing at a particular time.

Average Velocity

Average velocity over a time interval is defined as the overall displacement divided by the time elapsed. This concept helps in understanding the net rate of change in position, irrespective of the fluctuations in speed or direction during the interval.

Transcript

00:02 So we're going to be looking at problem 39 of chapter 2 of the physics fifth book.

00:06 The question says a position of a particle along the excesses dependent on the time is according to the equation x is a t squared minus b t cubed where x is in meters and t is in seconds.

00:18 For part a what s i units must a and b have and for the following letter numerical values have s i units being 3 1 respectively so for part b at what time does the particle reach as maximum x position part c what total path lengths is the particle cover in the first four seconds part d what's the displacement during the first four seconds part e what's the particle velocities at the end of each of the first four seconds part f what is the particle's acceleration at the end of each first four seconds and part g what's the average velocity for the time two to four so let's get cracking with part a where it asks you for the s units that a and b have.

01:03 Well we know that x is in meters so meters is equal to units of a times by second squared minus the units b is times by second squared.

01:13 We both of these have to be in meters overall so that means that ua is equal to meters over second squared and ub is equal to over seconds cubed.

01:25 For part b it tells us that let a be three and let b be and then it says what is the point in which the particle reaches maximum exposition? well we know that when the particle reaches max exposition something like this, then this is its max exposition here.

01:46 So it's gradient dx over dt equal to zero.

01:52 So dx over dt is equal to 2at minus 3bt squared, so that's just simple differentiation of this.

02:01 And we get put that equal to zero.

02:04 So subbing in a and b, we get 6t minus 3t squared is equal to zero.

02:12 And then we substitute out 3t from this whole thing to get 3t brackets 2 minus t is equal to zero.

02:22 So therefore we know t is either equal to 0 seconds or 2 seconds.

02:26 We discard 0 seconds as it's not useful answer to us...