At any instant the velocity of a particle of mass 500 g is (2t i + 3t^2 j) m s^-1. If the force acting on the particle at t = 1 s is ( i + x j) N, then the value of x will be:

Given: Mass of the particle is 500 g = 0.5 kg and velocity is v = (2t i + 3t^2 j). Find: The value of x in the force F = ( i + x j) N at t = 1 s. From the given velocity, acceleration is obtained by differentiating with respect to time: a = d vdt = (2 i + 6t j) At t = 1 s, a = (2 i + 6 j) m s^-2 Now use F = m a with m = 0.5 kg: F = 0.5 (2 i + 6 j) = ( i + 3 j) N Comparing with ( i + x j) N, we get x = 3. Therefore, the correct option is C. Given: v = (2t i + 3t^2 j) and m = 0.5 kg. Find: The coefficient x of the j-component of force at t = 1 s. Differentiate each component separately: ax = d(2t)dt = 2, ay = d(3t^2)dt = 6t So at t = 1 s, ax = 2, ay = 6 Multiply by mass to get force components: Fx = max = 0.5 2 = 1, Fy = may = 0.5 6 = 3 Hence, F = i + 3 j So the value of x is 3.

At any instant the velocity of a particle of mass 500 g is (2t i… — Solution | JEE Physics

MCQEasyJEE 2023Velocity & Acceleration

At any instant the velocity of a particle of mass $500 \, \text{g}$ is $\left(2t\hat{i} + 3t^2\hat{j}\right) \, \text{m s}^{-1}$ . If the force acting on the particle at $t = 1 \, \text{s}$ is $\left(\hat{i} + x\hat{j}\right) \, \text{N}$ , then the value of $x$ will be:

A
$2$
B
$6$
C
$3$
D
$4$

Answer

Correct answer:C

Step-by-step solution

Standard Method

Given: Mass of the particle is $500 \, \text{g} = 0.5 \, \text{kg}$ and velocity is $\vec{v} = \left(2t\hat{i} + 3t^2\hat{j}\right)$ .

Find: The value of $x$ in the force $\vec{F} = \left(\hat{i} + x\hat{j}\right) \, \text{N}$ at $t = 1 \, \text{s}$ .

From the given velocity, acceleration is obtained by differentiating with respect to time:

\vec{a} = \frac{d\vec{v}}{dt} = \left(2\hat{i} + 6t\hat{j}\right)

At $t = 1 \, \text{s}$ ,

\vec{a} = \left(2\hat{i} + 6\hat{j}\right) \, \text{m s}^{-2}

Now use $\vec{F} = m\vec{a}$ with $m = 0.5 \, \text{kg}$ :

\vec{F} = 0.5\left(2\hat{i} + 6\hat{j}\right) = \left(\hat{i} + 3\hat{j}\right) \, \text{N}

Comparing with $\left(\hat{i} + x\hat{j}\right) \, \text{N}$ , we get $x = 3$ .

Therefore, the correct option is C.

Using Components of Force

Given: $\vec{v} = \left(2t\hat{i} + 3t^2\hat{j}\right)$ and $m = 0.5 \, \text{kg}$ .

Find: The coefficient $x$ of the $\hat{j}$ -component of force at $t = 1 \, \text{s}$ .

Differentiate each component separately:

a_x = \frac{d(2t)}{dt} = 2, \qquad a_y = \frac{d(3t^2)}{dt} = 6t

So at $t = 1 \, \text{s}$ ,

a_x = 2, \qquad a_y = 6

Multiply by mass to get force components:

F_x = ma_x = 0.5 \times 2 = 1, \qquad F_y = ma_y = 0.5 \times 6 = 3

Hence,

\vec{F} = \hat{i} + 3\hat{j}

So the value of $x$ is $3$ .

Common mistakes

Differentiating the velocity incorrectly. The $\hat{j}$ -component is $3t^2$ , so its derivative is $6t$ , not $3t$ or $6t^2$ . Differentiate each component with respect to $t$ carefully.
Using mass as $500$ instead of converting $500 \, \text{g}$ to $0.5 \, \text{kg}$ . Force in SI units requires mass in kilograms, so always convert before applying $\vec{F} = m\vec{a}$ .
Comparing the whole vector without matching components. Since $\vec{F} = \left(\hat{i} + x\hat{j}\right)$ , the value of $x$ comes from the coefficient of $\hat{j}$ only. Match the $\hat{i}$ and $\hat{j}$ components separately.

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Answer

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