A particle is moving in the xy-plane and cros | vedclass

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(D) Given, the velocity of the particle is $\vec{v} = v_x \hat{i} + v_y \hat{j} = (2 \hat{i} + 2 \hat{j}) 	ext{ m s}^{-1}$.Thus, $v_x = \frac{dx}{dt}

Vedclass · Accepted Answer

$32$

Vedclass · Answer

(D) Given, the velocity of the particle is $\vec{v} = v_x \hat{i} + v_y \hat{j} = (2 \hat{i} + 2 \hat{j}) 	ext{ m s}^{-1}$.Thus, $v_x = \frac{dx}{dt} = 2 	ext{ m s}^{-1}$ and $v_y = \frac{dy}{dt} = 2 	ext{ m s}^{-1}$.The acceleration in the $x$-direction is $a_x = \frac{dv_x}{dt} = 0$ (since $v_x$ is constant).The acceleration in the $y$-direction is $a_y = \frac{dv_y}{dt} = a$.The trajectory is given by $y = 4x^2$.Differentiating with respect to time $t$, we get $\frac{dy}{dt} = 8x \frac{dx}{dt}$, which means $v_y = 8x v_x$.At the origin $(0,0)$, $x=0$. Substituting $v_y = 2$ and $v_x = 2$ into the equation $v_y = 8x v_x$ is not directly possible to find $x$ at the given velocity, but we differentiate $v_y = 8x v_x$ again with respect to $t$:$\frac{dv_y}{dt} = 8 \left( x \frac{dv_x}{dt} + v_x \frac{dx}{dt} ight)$.Substituting $a_y = a$, $a_x = 0$, $v_x = 2$, and $\frac{dx}{dt} = v_x = 2$:$a = 8(x \cdot 0 + 2 \cdot 2) = 8(4) = 32 	ext{ m s}^{-2}$.

Column $I$	Column $II$
$(A)$ Displacement	$(p)$ $8 \sin 2$
$(B)$ Distance	$(q)$ $4$
$(C)$ Average velocity	$(r)$ $2 \sin 2$
$(D)$ Average acceleration	$(s)$ $4 \sin 2$

$A$ particle is moving in the $xy$-plane and crosses the origin at time $t=0$. The equation of motion of the particle is $y=4x^2$. If the velocity of the particle is $\vec{v}=(2\hat{i}+2\hat{j}) \text{ m s}^{-1}$ and acceleration is $\vec{a}=(a\hat{j}) \text{ m s}^{-2}$, then the magnitude of $a$ is

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