Question

A particle moves in a straight line, according to the law \[x=4a[t+a\sin (\dfrac{t}{a})]\], where \[x\] is its position in meters, \[t\] in sec. & \[a\] is some constants, then velocity is zero at
A) \[x=4{{a}^{2}}\pi meters\]
B) \[t=\pi \sec .\]
C) \[t=0\sec .\]
D) \[none\]

Answer 1

Hint: Velocity is the rate of change of displacement. To get velocity the equation of displacement must be differentiated with respect to time. Then solve as per the given question.

Complete step by step answer:
The given equation is for displacement and we are required to find the time when velocity is zero.
Since velocity is the rate of change of displacement hence we will differentiate the given equation.
$\Rightarrow x=4a[t+a\sin (\dfrac{t}{a})]$
Here \[x\] is the displacement
\[a\] is a constant
\[t\] is the time in seconds

Let’s first simplify this equation,
\[\Rightarrow x=4at+4{{a}^{2}}\sin (\dfrac{t}{a})\]
Now differentiating this equation with respect to time we will get equation of velocity since $v=\dfrac{dx}{dt}$
\[\Rightarrow \dfrac{dx}{dt}=4a+4{{a}^{2}}(\cos (\dfrac{t}{a}))(\dfrac{1}{a})\]
Here \[\dfrac{d(4at)}{dt}=4a\] since \[a\] is constant as given in the question.
And it is also known from basic differentiation rules that \[\dfrac{d(\sin \theta )}{d\theta }=\cos \theta \]
Also \[\dfrac{d(\sin a\theta )}{d\theta }=(\cos a\theta )(\dfrac{da\theta }{d\theta })\]
We finally get
\[\Rightarrow v=\dfrac{dx}{dt}=4a+4a\cos (\dfrac{t}{a})\]
That is equation of velocity is given as
\[\Rightarrow v=4a+4a\cos (\dfrac{t}{a})\]
We need to calculate time for which the velocity is zero, hence we will put \[v=0\] in the above equation
$\Rightarrow 0=4a+4a\cos (\dfrac{t}{a})$
On solving further we get,
$\Rightarrow 0=4a[1+\cos \dfrac{t}{a})]$
$\Rightarrow 0=1+\cos (\dfrac{t}{a})$
$\Rightarrow -1=\cos (\dfrac{t}{a})$
Now this equation is of the form, $\cos \theta =-1$
We need to find $\theta $ such that $\cos \theta =-1$, let’s check the graph of $\cos \theta $
The value of $\cos \theta $ is equal to $-1$ when the value of $\theta $ is $\pi $.
Therefore, $\cos (\dfrac{t}{a})=-1$
$\Rightarrow \dfrac{t}{a}=\pi $
$\Rightarrow t=a\pi $
Therefore the velocity is zero at $t=a\pi $ which is not given in the option.
Hence the correct option is $(D)$ none of these

Note: If we again differentiate the equation of velocity with respect to time, we will get an equation for acceleration. Do remember that there can be infinitely many solutions as the graph of \[\cos \theta \] is periodic in nature. It repeats after an interval of \[2\pi \]. This implies that the velocity will again be zero for \[t=\pi a+2\pi a\], \[t=\pi a+4\pi a\] and so on.