The correct option is D 1.5 s
Given,
mass of the particle, m=10−2 kg
Force acting on the particle,
F=−K2x2...(1)
We know that force acting on the moving particle,
F=ma...(2)
Equating (1) and (2)
⇒ma=−K2x2
Substituting the values of m and K in the above equation,
⇒10−2×a=−10−22x2
⇒a=−12x2
⇒vdvdx=−12x2 [∵a=vdvdx]
Rearranging and integrating the equation, as x=1 m,v=0 m\s,
⇒v∫0vdv=−12x∫1−x−2dx
⇒12[v2]v0=+12[x−1]x1
⇒v2=[1x−1]=1−xx
⇒v=±√1−xx
Initially the particle was moving in +x-direction at x=1. When the particle is at x=0.5, obviously its velocity will be in -x-direction. The force acting in -x-direction first decreases the speed of the particle, bring it momentarily at rest and then changes the direction of motion of the particle.
Thus, we should consider the negative velocity.
⇒v=dxdt=−√1−xx
⇒0.25∫1√x1−xdx=−t∫0dt
On taking x=sin2θ, we get dx=2sinθcosθdθ.
And limits as when x=1,θ=90∘
and when x=0.25,θ=30∘.
30∘∫90∘√sin2θ1−sin2θ×2sinθcosθdθ=−t∫0dt
30∘∫90∘2sin2θdθ=−t∫0dt
30∘∫90∘(1−cos2θ)=−t∫0dt
⇒[θ−sin2θ2]30∘90∘=−[t]t0
⇒−t=30∘−sin60∘2−90∘+sin180∘2
⇒−t=π6−√34−π2+0
⇒−t=−1.48
∴t=1.48 s≈1.5 s