Motion in a plane

The velocity of a particle is given as $\vec v=-x\hat i+2y\hat j-z\hat k$ m/s. The magnitude of acceleration at point $(1,2,4)$ is ________ m/s².

At $t=0$, a body of mass 100 g starts moving under the influence of a force $\left(5\hat i+10\hat j\right)$ N. After 2 s its position is $\left(2x\hat i+5y\hat j\right)$ m. The ratio $x:y$ is ________.

Two identical bodies A and B of equal masses have initial velocities $\vec v_1=4\hat i$ m/s and $\vec v_2=4\hat j$ m/s respectively. The body A has acceleration $\vec a_1=6\hat i+6\hat j$ m/s² while the acceleration of the other body B is zero. The centre of mass of the two bodies moves in ________ path.

A particle of mass $m$ is moving in the $xy$-plane such that its velocity at a point $(x, y)$ is given as $\vec v = \alpha(y\hat x + 2x\hat y)$, where $\alpha$ is a non-zero constant. What is the force $\vec F$ acting on the particle?

A force on an object of mass 100 g is $\left(10\hat i+5\hat j\right)$ N. The position of that object at t = 2 s is $\left(a\hat i+b\hat j\right)$ m after starting from rest. The value of $\dfrac ab$ will be ______.

Motion of a particle in $x$-$y$ plane is described by a set of following equations $x=4\sin\left(\dfrac\pi2-\omega t\right)$ m and $y=4\sin(\omega t)$ m. The path of the particle will be :

At $t=0$, truck, starting from rest, moves in the positive $x$-direction at uniform acceleration of 5 ms$^{-2}$. At $t=20$ s, a ball is released from the top of the truck. The ball strikes the ground in 1 s after the release. The velocity of the ball, when it strikes the ground, will be :

(Given $g=10$ ms$^{-2}$)

A person starts his journey from centre 'O' of the park and comes back to the same position following path OPQO as shown in the figure. The radius of path taken by the person is 200 m and he takes 3 min 58 sec to complete his journey. The average speed of the person is ______ ms$^{-1}$.

(take $\pi=3.14$)

A person moved from A to B on a circular path as shown in figure. If the distance travelled by him is 60 m, then the magnitude of displacement would be :

(Given $\cos135°=-0.7$)

At time t = 0 a particle starts travelling from a height $7\hat z$ cm in a plane keeping z coordinate constant. At any instant of time it's position along the $\hat x$ and $\hat y$ directions are defined as 3t and 5t$^3$ respectively. At t = 1 s acceleration of the particle will be

Starting at time $t = 0$ from the origin with speed $1\ \mathrm{m\,s^{-1}}$, a particle follows a two-dimensional trajectory in the $x$-$y$ plane so that its coordinates are related by the equation $y = \dfrac{x^2}{2}$. The $x$ and $y$ components of its acceleration are denoted by $a_x$ and $a_y$, respectively. Then

A ball is thrown from ground at an angle $\theta$ with horizontal and with an initial speed $u_0$. For the resulting projectile motion, the magnitude of average velocity of the ball up to the point when it hits the ground for the first time is $V_1$. After hitting the ground, the ball rebounds at the same angle $\theta$ but with a reduced speed of $u_0/\alpha$. Its motion continues for a long time as shown in figure. If the magnitude of average velocity of the ball for entire duration of motion is $0.8V_1$, the value of $\alpha$ is ______.

Consider a body of mass 1.0 kg at rest at the origin at time $t = 0$. A force $\vec F = (\alpha t\,\hat i + \beta\,\hat j)$ is applied on the body, where $\alpha = 1.0\ \mathrm{N\,s^{-1}}$ and $\beta = 1.0$ N. The torque acting on the body about the origin at time $t = 1.0$ s is $\vec\tau$. Which of the following statements is (are) true?

The position vector $\vec r$ of a particle of mass $m$ is given by the following equation
$$\vec r(t)=\alpha t^3\,\hat i+\beta t^2\,\hat j,$$
where $\alpha=10/3\ \mathrm{m\,s^{-3}}$, $\beta=5\ \mathrm{m\,s^{-2}}$ and $m=0.1$ kg. At $t=1$ s, which of the following statement(s) is(are) true about the particle?

$x$ and $y$ displacements of a particle are given as $x(t)=a\sin\omega t$ and $y(t)=a\sin2\omega t$. Its trajectory will look like :