Question

At $t=0$, an object of mass $m$is at rest at $x=0$on a horizontal, frictionless surface. Starting at $t=0$, a horizontal forcé $F_x=F_0{e}^{-t/T}$is exerted on the object.

a. Find and graph an expression for the object's velocity at an arbitrary later time $t$.

b. What is the object's velocity after a very long time has elapsed?

Short answer

Part(a): The expression of particle velocity at any time $t$is $v\left(t\right)=\frac{F_{0}T}{m}[1-e^{-t/T}]$.

Part (b): Objects velocity after a very long time is$\frac{F_{0}T}{m}$.

Step-by-step solution

Part (a) Step 1: Given.

Force on the particle, $F_x=F_o{e}^{-t/T}$.

Velocity at time $t=0$.

Position at time$v=0$.

Part (b): Step 2: Formula used.

Force on a object is given by the equation:

$F=ma$

Here,

$m$is the mass.

$a$is the acceleration.

Part (b): Step 3: Calculation

Force on the particle is given as:

$F_x=F_o{e}^{-t/T}\dots \dots \text{(1)}$

Force on an object is given by the equation

$F=ma\dots ..\left(2\right)$

On comparing Eq. $\left\{1\right\}$and Eq.{$2$}

role="math" localid="1647718262854" $\begin{array}{r}ma=F_o{e}^{-t/T}\\ a=\frac{F_s{e}^{-t/T}}{m}\end{array}$

The velocity of a particle at a time $t$is obtain by integrating the above equation,

role="math" localid="1647718224750" $\begin{array}{r}\int adt=\int \frac{F_o{e}^{-t/T}}{m}dt\\ \int \frac{dv}{d}dt=\frac{F_0}{w_0}\left[-T{e}^{-t/T}\right]\dots \dots \text{(3)}\end{array}$

$v=-\frac{F_{0}T}{m}\left[e^{-t/T}\right]+C$

Part (b) Step 4: Calculation.

Here,

$C$is the arbitrary constant of integration

Apply the boundary condition that at time $t=0$the velocity of the particle is $0$.

Plugging the values in the above equation

$\begin{array}{rl}& v=-\frac{F_{0}T}{m}\left[e^{-t/T}\right]+C\\ & 0=-\frac{F_{0}T}{m}\left(e^0\right)+C\\ & C=\frac{F_{0}T}{m}\end{array}$

Substitute the value of $C$in Eq. ($3$

$\begin{array}{rl}v=& -\frac{F_{0}T}{m}\left[e^{-t/T}\right]+\frac{F_{0}T}{m}\\ & =\frac{F_{0}T}{m}[1-e^{-t/T}]\end{array}$

Part (b) Step 5: GIven.

The expression of particle velocity at any time $t$is$v\left(t\right)=\frac{F_{0}T}{m}[1-e^{-t/T}]$.

Part (b): Step 6; Calculation.

Expression of particle velocity at any time$t$ is:

$v\left(t\right)=\frac{F_{0}T}{m}[1-e^{-t/T}]$

The velocity of a particle is an exponent function of time, therefore as the time elapsed the magnitude of exponent term will be $0$.

And the velocity of the particle is given as:

$\begin{array}{c}v\left(t\right)=\frac{F_{0}T}{m}[1-e^{-\propto /T}]\\ =\frac{F_{0}T}{m}[1-0]\\ =\frac{F_{0}T}{m}\end{array}$

Conclusions.

Part (a): Expression of particle velocity at any time$t$is $v\left(t\right)=\frac{F_{0}T}{m}[1-e^{-t/T}]$.

Part (b): Objects velocity after a very long time is$\frac{F_{0}T}{m}$.