Question

$y(x,t)=\frac{0.8}{\left[\right(4x+5t{)}^2+5]}$ represents a moving pulse where $x$ and $y$ are in metre and $t$ in second. Then

• A. Pulse is moving in positive $x-$direction.
• B. In $2$ second, it will travel a distance of $2.5m$
• C. Its maximum displacement is $0.16m$
• D. It is a symmetric pulse

Answer 1

Answer: (B), (C), (D)

The correct options are
B In $2$ second, it will travel a distance of $2.5m$
C Its maximum displacement is $0.16m$
D It is a symmetric pulse

The shape of pulse at $x=0$ and $t=0$ would be,


$y(0,0)=\frac{0.8}{5}=0.16m$
From the figure, it is clear that $y_{max}=0.16m$

Pulse will be symmetric (symmetry is checked about $y_{max})$ if at $t=0$,
$y\left(x\right)=y(-x)$

From the given equation,
$y\left(x\right)=\frac{0.8}{16x^2+5}att=0$
and
$y(-x)=\frac{0.8}{16x^2+5}$
(or) $y\left(x\right)=y(-x)$
Therefore, pulse is symmetric.

For speed of pulse:

If equation of a wave pulse is $y=f[ax\pm bt]$
The speed of wave is $\left(\frac{b}{a}\right)$ in negative X-direction for $y=f[ax+bt]$ and positive X direction for $y=f[ax-bt]$.
Comparing this with given equation, we can find that speed of wave is $\frac{5}{4}=1.25m{s}^{-1}$ and it is traveling in negative X-direction.
$\Rightarrow$ wave pulse travels $2.5m$ in $2s$