A body of mass 500 g moves along x-axis such that its velocity va… — Solution | JEE Physics

MCQEasyJEE 2023Velocity & Acceleration

A body of mass $500 \, \text{g}$ moves along x-axis such that its velocity varies with displacement $x$ according to the calculation $v = 10\sqrt{x} \, \text{m/s}$ the force acting on the body is :-

A
$25 \, \text{N}$
B
$5 \, \text{N}$
C
$166 \, \text{N}$
D
$125 \, \text{N}$

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given:

Mass $m = 500 \, \text{g} = 0.5 \, \text{kg}$
Velocity $v = 10\sqrt{x} \, \text{m/s}$

Find: The force acting on the body.

For motion where velocity is given as a function of displacement, acceleration is

a = \frac{dv}{dt} = \frac{dv}{dx} \cdot \frac{dx}{dt} = v\frac{dv}{dx}

Differentiate $v = 10\sqrt{x}$ with respect to $x$ :

\frac{dv}{dx} = \frac{d}{dx}(10\sqrt{x}) = 10 \cdot \frac{1}{2\sqrt{x}} = \frac{5}{\sqrt{x}}

Now substitute in $a = v\frac{dv}{dx}$ :

a = 10\sqrt{x} \cdot \frac{5}{\sqrt{x}} = 50 \, \text{m/s}^2

Using Newton's second law,

F = ma

Substitute $m = 0.5 \, \text{kg}$ and $a = 50 \, \text{m/s}^2$ :

F = 0.5 \cdot 50 = 25 \, \text{N}

Therefore, the force acting on the body is $25 \, \text{N}$ . The correct option is A.

Common mistakes

Using $a = \frac{dv}{dx}$ directly is incorrect because acceleration is rate of change of velocity with respect to time, not displacement. Use $a = \frac{dv}{dt} = v\frac{dv}{dx}$ instead.
Not converting mass from $500 \, \text{g}$ to $0.5 \, \text{kg}$ gives the wrong force. Always convert to SI units before applying $F = ma$ .
Differentiating $10\sqrt{x}$ incorrectly can lead to a variable-dependent answer. Since $\frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}}$ , we get $\frac{dv}{dx} = \frac{5}{\sqrt{x}}$ , which cancels with $\sqrt{x}$ in $v\frac{dv}{dx}$ .

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