The position vector of a moving particle at time $t$ is $r =3 \hat{ i }+4 t \hat{ j }-t \hat{ k }$. Its displacement during the time interval $t=1 s$ to $t=3 s$ is

A particle is moving in a circular path. The acceleration and momentum of the particle at a certain moment are $\vec a = (4\hat i + 3\hat j)\ m/s^2$ and $\vec p = (8\hat i - 6\hat j)\ kg-m/s$ . The motion of the particle is

The initial position of an object at rest is given by $3 \hat{i}-8 \hat{j}$ It moves with constant acceleration and reaches to the position $2 \hat{i}+4 \hat{j}$ after $4 \,s$. What is its acceleration?

A particle starts from origin at $t=0$ with a velocity $5.0 \hat{ i }\; m / s$ and moves in $x-y$ plane under action of a force which produces a constant acceleration of $(3.0 \hat{ i }+2.0 \hat{ j })\; m / s ^{2} .$

$(a)$ What is the $y$ -coordinate of the particle at the instant its $x$ -coordinate is $84 \;m$ ?

$(b)$ What is the speed of the particle at this time?

Tangential acceleration of a particle moving in a circle of radius $1$ $m$ varies with time $t$ as (initial velocity of particle is zero). Time after which total acceleration of particle makes and angle of $30^o$ with radial acceleration is

A particle is moving along the $x-$axis with its coordinate with the time '$t$' given be $\mathrm{x}(\mathrm{t})=10+8 \mathrm{t}-3 \mathrm{t}^{2} .$ Another particle is moving the $y-$axis with its coordinate as a function of time given by $\mathrm{y}(\mathrm{t})=5-8 \mathrm{t}^{3} .$ At $\mathrm{t}=1\; \mathrm{s},$ the speed of the second particle as measured in the frame of the first particle is given as $\sqrt{\mathrm{v}} .$ Then $\mathrm{v}$ (in $\mathrm{m} / \mathrm{s})$ is