A particle is moving with velocity $\vec { v } = K ( y \hat { i } + x \hat { j } )$, where $K$ is a constant. The general equation for its path is (1) $y = x ^ { 2 } +$ constant (2) $y ^ { 2 } = x +$ constant (3) $x y =$ constant (4) $y ^ { 2 } = x ^ { 2 } +$ constant
A small particle of mass $m$ is projected at an angle $\theta$ with the x-axis with an initial velocity $\mathrm { v } _ { 0 }$ in the $\mathrm { x } - \mathrm { y }$ plane as shown in the figure. At a time $t < \frac { v _ { 0 } \sin \theta } { g }$, the angular momentum of the particle is where $\hat { \mathrm { i } } , \hat { \mathrm { j } }$ and $\hat { \mathrm { k } }$ are unit vectors along $\mathrm { x } , \mathrm { y }$ and z-axis respectively. (1) $- \mathrm { mgv } _ { 0 } \mathrm { t } ^ { 2 } \cos \theta \hat { \mathrm { j } }$ (2) $\mathrm { mgv } _ { 0 } t \cos \theta \hat { \mathrm { k } }$ (3) $- \frac { 1 } { 2 } m g v _ { 0 } t ^ { 2 } \cos \theta \hat { k }$ (4) $\frac { 1 } { 2 } m g v _ { 0 } t ^ { 2 } \cos \theta \hat { i }$
Two fixed frictionless inclined plane making an angle $30 ^ { \circ }$ and $60 ^ { \circ }$ with the vertical are shown in the figure. Two block $A$ and $B$ are placed on the two planes. What is the relative vertical acceleration of $A$ with respect to $B$? (1) $4.9 \mathrm {~ms} ^ { - 2 }$ in horizontal direction (2) $9.8 \mathrm {~ms} ^ { - 2 }$ in vertical direction (3) zero (4) $4.9 \mathrm {~ms} ^ { - 2 }$ in vertical direction
The figure shows the position-time $(x-t)$ graph of one-dimensional motion of a body of mass 0.4 kg. The magnitude of each impulse is (1) 0.4 Ns (2) 0.8 Ns (3) 1.6 Ns (4) 0.2 Ns
A point P moves in counter-clockwise direction on a circular path as shown in the figure. The movement of 'P' is such that it sweeps out a length $s = t ^ { 3 } + 5$, where $s$ is in metres and $t$ is in seconds. The radius of the path is 20 m. The acceleration of 'P' when $t = 2 \mathrm {~s}$ is nearly (1) $13 \mathrm {~m} / \mathrm { s } ^ { 2 }$ (2) $12 \mathrm {~m} / \mathrm { s } ^ { 2 }$ (3) $7.2 \mathrm {~m} / \mathrm { s } ^ { 2 }$ (4) $14 \mathrm {~m} / \mathrm { s } ^ { 2 }$
For a particle in uniform circular motion the acceleration $\vec { a }$ at a point $P ( R , \theta )$ on the circle of radius R is (here $\theta$ is measured from the $x$-axis) (1) $- \frac { v ^ { 2 } } { R } \cos \theta \hat { i } + \frac { v ^ { 2 } } { R } \sin \theta \hat { j }$ (2) $- \frac { v ^ { 2 } } { R } \sin \theta \hat { i } + \frac { v ^ { 2 } } { R } \cos \theta \hat { j }$ (3) $- \frac { v ^ { 2 } } { R } \cos \theta \hat { i } - \frac { v ^ { 2 } } { R } \sin \theta \hat { j }$ (4) $\frac { v ^ { 2 } } { R } \hat { i } + \frac { v ^ { 2 } } { R } \hat { j }$