If the displacement, velocity and acceleration of a particle at time, t be x, v and f respectively, then which one is true?

Question

If the displacement, velocity and acceleration of a particle at time, t be x, v and f respectively, then which one is true? (A) f = v3 (d2t/dx2) (B) f = -v3 (d2t/dx2) (C) f = v2 (d2t/dx2) (D) f = -v2 (d2t/dx2)

Tardigrade Admin · Accepted Answer

(d2 t/d x2)=(d((d t/d x))/d x)=(d((1/v))/d x) =-(1/v2) (d v/d t) × (1/v) ⇒ f=-v3 (d2 t/d x2)

Q. If the displacement, velocity and acceleration of a particle at time, $t$ be $x, v$ and f respectively, then which one is true?

$f = v^{3} \frac{d ^{2} t}{d x ^{2}}$

$f = - v^{3} \frac{d ^{2} t}{d x ^{2}}$

$f = v^{2} \frac{d ^{2} t}{d x ^{2}}$

$f = - v^{2} \frac{d ^{2} t}{d x ^{2}}$

$\frac{d ^{2} t}{d x ^{2}} = \frac{d ( \frac{d t}{d x} )}{d x} = \frac{d ( \frac{1}{v} )}{d x}$
$= - \frac{1}{v ^{2}} \frac{d v}{d t} \times \frac{1}{v}$
$\Rightarrow f = - v^{3} \frac{d ^{2} t}{d x ^{2}}$

Q. If the displacement, velocity and acceleration of a particle at time, t be x,v and f respectively, then which one is true?

f=v3dx2d2t​

f=−v3dx2d2t​

f=v2dx2d2t​

f=−v2dx2d2t​