Motion in a Plane (2-D Kinematics)

The position vector of a moving body at any instant of time is given as$\vec{r} = \left(5t^{2}\hat{i} - 5t\hat{j}\right)\text{ m}$. The magnitude and direction of velocity at $t = 2\text{ s}$ is,

Position of an ant ($\text{S}$ in metres) moving in $\text{Y-Z}$ plane is given by $S = 2t^{2}\hat{j} + 5\hat{k}$ (where $t$ is in second). The magnitude and direction of velocity of the ant at $t = 1\text{ s}$ will be:

The co-ordinates of a particle moving in $x\text{-}y$ plane are given by: $x = 2 + 4t,\ y = 3t + 8t^{2}$. The motion of the particle is:

A particle starts from origin at $t = 0$ with a velocity $5\hat{i}\text{ m/s}$ and moves in $x - y$ plane under action of a force which produces a constant acceleration of $(3\hat{i} + 2\hat{j})\text{ m/s}^{2}$. If the $x$-coordinate of the particle at that instant is $84\text{ m}$, then the speed of the particle at this time is $\sqrt{\alpha}\text{ m/s}$. The value of $\alpha$ is _________.

At time $t = 0$ a particle starts travelling from a height $7\hat{z}\text{ cm}$ in a plane keeping z coordinate constant. At any instant of time its position along the $\hat{x}$ and $\hat{y}$ directions are defined as $3t$ and $5t^{3}$ respectively. At t = 1s acceleration of the particle will be

A mosquito is moving with a velocity$\vec{v} = 0.5t^{2}\hat{i} + 3t\hat{j} + 9\hat{k}$ m/s and accelerating in uniform conditions. What will be the direction of mosquito after 2 s?

A particle moves such that its position vector$\vec{r}(t) = \cos \omega t\, \hat{i} + \sin \omega t\, \hat{j}$ where $\omega$ is a constant and t is time. Then which of the following statements is true for the velocity$\vec{v}(t)$ and acceleration$\vec{a}(t)$ of the particle:

A particle is moving along the x-axis with its coordinate with the time 't' given be x(t) = 10 + 8t – 3t2. Another particle is moving the y-axis with its coordinate as a function of time given by y(t) = 5 – 8t3.

At t = 1s, the speed of the second particle as measured in the frame of the first particle is given as $\sqrt{v}$. Then v (in m/s) is ______.

A balloon is moving up in air vertically above a point A on the ground. When it is at a height h1, a girl standing at a distance d (point B) from A (see figure) sees it at an angle 45o with respect to the vertical. When the balloon climbs up a further height h2, it is seen at an angle 60o with respect to the vertical if the girl moves further by a distance 2.464 d (point C). Then the height h2 is (given tan 30o = 0.5774)

Starting from the origin at time t = 0, with initial velocity 5$\hat{j}$ ms-1, a particle moves in the x-y plane with a constant acceleration of $\left(10\hat{i} + 4\hat{j}\right)$ ms-2. At time t, its coordinates are (20 m, y0 m). The values of t and y0 are, respectively:

A particle starts from the origin at t = 0 with an

initial velocity of 3.0 $\hat{i}$ m/s and moves in the

x-y plane with a constant acceleration $\left(6\hat{i} + 4\hat{j}\right)$ m/s2. The x-coordinate of the particle at the instant when its y-coordinate is 32 m is D meters. The value of D is :-

The position co-ordinates of a particle moving in a 3-D coordinate system is given by x = a cos$\omega$t

y = a sin$\omega$t and

z = a$\omega$t

The speed of the particle is:

A particle is moving with a velocity

$\vec{v} = K(y\hat{i} + x\hat{j}),$ where K is a constant.

The general equation for its path is:

The x and y coordinates of the particle at any time are x = 5t $-$ 2t2 and y = 10t respectively, where x and y are in metres and t in seconds. The acceleration of the particle at t = 2 s is

A particle moves so that its position vector is given by $\vec{r} = \cos \omega t\, \hat{x} + \sin \omega t\, \hat{y},$ where $\omega$ is a constant. Which of the following is true?

The position vector of a particle $\vec{R}$as a function of time is given by $\vec{R}$ = 4sin(2$\pi t)\hat{i}$ + 4cos(2$\pi t)\hat{j}$. Where R is in meters, t is in seconds and $\hat{i}$ and $\hat{j}$ denote unit vectors along x-and y-directions, respectively. Which one of the following statements is wrong for the motion of particle?

A particle is moving such that its position coordinates (x, y) are (2 m, 3 m) at time t = 0, (6 m, 7 m) at time t = 2 s and (13 m, 14 m) at time t = 5 s.

Average velocity vector $\left(\vec{v}_{av}\right)$ from t = 0 to t = 5 s is

A particle has initial velocity $\left(2\vec{i} + 3\vec{j}\right)$ and acceleration $\left(0.3\vec{i} + 0.2\vec{j}\right)$. The magnitude of velocity after 10 seconds will be

A body is moving with velocity 30 m/s towards east. After 10 seconds its velocity becomes 40 m/s towards north. The average acceleration of the body is

A particle moves in x-y plane according to rule x = asin$\omega$t and y = acos$\omega$t. The particle follows

A particle has initial velocity $\left(3\hat{i} + 4\hat{j}\right)$ and has acceleration $\left(0.4\hat{i} + 0.3\hat{j}\right).$ Its speed after 10 s is

A particle is moving with velocity$\vec{v} = k\left(y\hat{i} + x\hat{j}\right)$, where K is a constant. The general equation for its path is

A particle has an initial velocity $3\hat{i} + 4\hat{j}$ and an acceleration of $0.4\hat{i} + 0.3\hat{j}$. Its speed after 10 s is:

A particle starting from the origin (0, 0) moves in a straight line in the (x, y) plane. Its coordinates at a later time are $\left(\sqrt{3}, 3\right)$. The path of the particle makes with the x-axis an angle of

A particle is moving eastwards with a velocity of 5 m/s. In 10 seconds the velocity changes to 5 m/s northwards. The average acceleration in this time is

Two particles $A$ and $B$ are connected by a rigid rod $AB$. The rod slides along perpendicular rails as shown here. The velocity of $A$ to the right is $10\text{ m/s}$. What is the velocity of $B$ when angle $\alpha = 60^\circ$?

The position vector of a particle is $\vec{r} = (a \cos \omega t)\hat{i} + (a \sin \omega t)\hat{j}$. The velocity of the particle is

A bus is moving on a straight road towards north with a uniform speed of 50 km/hour then it turns left through $90^\circ$. If the speed remains unchanged after turning, the increase in the velocity of bus in the turning process is