Question

A particle starting from rest moves in a straight line with acceleration proportional to x^2, (where x i displacement). The gain of kinetic energy for any displacement is proportional to :
1) x^3
2)x^1/2
3)x^2/3
4)x^2

Answer 1

Acceleration is proportional to x²
e.g., a = Kx²
Here , a is the acceleration and K is proportionality constant
vdv/dx = Kx²
⇒∫vdv = K∫x²dx
⇒ v²/2 = Kx³/3
⇒v² = 2Kx²/3 ----------(1)

we know,
kinetic energy = 1/2 mv²
= 1/2 m (2Kx³/3) = mKx³/3 [ from equation (1)
Here it is clear that kinetic energy is directly proportional to x³.
So, correct option ( 1 )

Answer 2

》》Here , it is said that "Acceleration (a) is proportional to Displacement ($x^{2}$).

So , by removing proportionality

$a = kx^{2}$ , where 'k' is a proportionality constant

We can write $a = \frac{v×dv}{x}$....( you can check by seeing unit using dimensional formula)

$\frac{v×dv}{dx} = kx^{2}$

Multiply both side by dx
$v×dv = kx^{2} × dx$

Integrating both sides

$∫v×dv = ∫kx^{2} × dx$

$\frac{v^{2}}{2} = \frac{x^{3} }{3}$........ Since ∫x×dx =$\frac{x^(n+1) }{n+1}$

$v^{2 }= \frac{2x^{3}}{3}$

As we got $v^{2}$ we can substitute its value in Equation of Kinetic energy.

Kinetic Energy = $\frac{1}{2} mv^{2}$

Substituting value of $v^{2}$

Kinetic Energy = $\frac{1}{2} m× \frac{2x^{3}}{3}$

Kinetic Energy = $\frac{mx^{3}}{3}$

Take $\frac{m}{3} = K$

Therefore,

Kinetic Energy = $Kx^{3}$

So , Kinetic energy is directly proportional to $x^{3}$

I.e your answer OPTION [1]