Q16P - Fundamentals Of Physics

Question

The position function $\mathbf{x}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$ of a particle moving along an x axis is $\mathbf{x}\mathbf{=}\mathbf{4}\mathbf{-}\mathbf{6}{\mathbf{t}}^{\mathbf{2}}$ , with x in metres and t in seconds. (a) At what time does the particle momentarily stop? (b) Where does the particle momentarily stop? At what (c) negative time and (d) positive time does the particle pass through the origin? (e) Graph x vs t for range $\mathbf{-}\mathbf{5}\mathbf{ }\mathbf{sec}$   to $\mathbf{+}\mathbf{5}\mathbf{ }\mathbf{sec}$. (f) To shift the curve rightward on the graph, should we include the term   $\mathbf{+}\mathbf{20}\mathbf{t}$ or $\mathbf{-}\mathbf{20}\mathbf{t}$ in ? (g) Does that inclusion increase or decrease the value of x at which the particle momentarily stops?

Step-by-Step Solution

Answer

1. Particle stops at $\mathrm{t}=0$
2. Position of particle when it stops is $4.0\mathrm{m}$.
3. Negative time when particle passes through origin is $\mathrm{t}=-0.82 \mathrm{s}$.
4. Positive time when particle passes through origin is $\mathrm{t}=+0.82 \mathrm{s}$
5. $v=0$ At larger values of $\mathrm{x}$.

Step 1: Given information

$\mathrm{x}=4-6{\mathrm{t}}^2$

Step 2: To understand the concept

The problem involves differentiation of the quantity. Here the functional notation can be used for differentiation. To initiate, vand acan be calculated from the given equation by taking its derivative. By plugging the value of $\mathrm{t}=0$  in equation of position to find position at this time. At $\mathrm{x}\left(\mathrm{t}\right)=0$  solve quadratic equation for $\mathrm{t}$. The graph for $\mathrm{x}$  vs $\mathrm{t}$ can be drawn. Also by adding $20\mathrm{t}$we can draw shifted graph.

Formula:

$\mathrm{v}\left(\mathrm{t}\right)=\frac{\mathrm{dx}\left(\mathrm{t}\right)}{\mathrm{dt}}$

Step 3: Find the time when the particle momentarily stops

Find velocity and acceleration from given position by taking derivative of it

$$\begin{gathered} \mathrm{v}\left(\mathrm{t}\right)=\frac{\mathrm{dx}\left(\mathrm{t}\right)}{\mathrm{dt}} \\ =-12\mathrm{t} \end{gathered}$$

$$\begin{gathered} a\left(\mathrm{t}\right)=\frac{dv\left(\mathrm{t}\right)}{\mathrm{dt}} \\ =-12 \end{gathered}$$

It is found that at $\mathrm{t}=0$ the velocity is $0$.

Step 4: To find where does the particle momentarily stop

$\mathrm{x}=4-6{\mathrm{t}}^2$

At $\mathrm{t}=0$ the position of particle is $\mathrm{x}\left(0\right)=4.0 \mathrm{m}$

Step 5: To find negative and positive time the particle pass through the origin

Solve $\mathrm{x}\left(\mathrm{t}\right)=0$

$0=4.0-6.0{\mathrm{t}}^2$

By solving this quadratic equation for the negative time $\mathrm{t}=-0.82 \mathrm{s}$

Step 6: To plot the graph x vs t for range to and to find what to shift the curve rightward on the graph

Alsothe positive time from above quadratic equation  $\mathrm{t}=+0.82 \mathrm{s}$

Both the graphs (on the left) as well as the shifted graph can be drawn. In both graphs time limits are $-3\le \mathrm{t}\le 3.$

The graph 2 can be drawn by adding $20\mathrm{t}$ in $\mathrm{x}\left(\mathrm{t}\right)$ expression.

The slopes of graphs as zero, shift causes $\mathrm{y}=0$ at larger values of. $\mathrm{x}$