A particle moves in a straight line. If u is the velocity at a distance x from a fixed point on the line and v2=a-bx2,where a and b arc constants, then:(i) the motion continues along the positive r-direction only(ii) the motion is simple harmonic(iii) the particle oscillates with a frequency equal to b2π(iv) the total energy of the particle is ma

A particle moves in a straight line. If u is the velocity at a distance x from a fixed point on the line and $v^{2} = a - {bx}^{2},$ where a and b arc constants, then:
(i) the motion continues along the positive r-direction only
(ii) the motion is simple harmonic
(iii) the particle oscillates with a frequency equal to $\frac{\sqrt{b}}{2 π}$
(iv) the total energy of the particle is ma

A
(i, ii)
B
(ii, iii)
C
(iii, iv)
D
(ii, iv)

Solution:

$v^{2} = a - {bx}^{2} \Rightarrow 2 v dv = 0 - b (2 x dx) \Rightarrow v \frac{dv}{dx} = - bx \Rightarrow \frac{d^{2} x}{{dx}^{2}} + bx = 0 \Rightarrow motion is SHM and ω^{2} = b \Rightarrow f = \frac{ω}{2 π} = \frac{\sqrt{2}}{2 π}$

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