Kinematics

Consider a large disk of radius $R$ and two smaller disks, each of radius $r=R/50$, lying on its circumference, as shown in the figure. The smaller disks are initially in contact with each other, with an angular separation $\Delta\theta$ between their centers. They are made to roll without slipping in opposite directions, with constant angular velocities $\omega$ and $2\omega$ while the large disk is held stationary. The time $\tau$ at which the smaller disks are again in contact is:
[Use $\sin(\Delta\theta)=\Delta\theta$ and ignore gravity.]

A particle is thrown with a speed $v$ from a point $O$ at an angle $\theta$ with the horizontal plane such that it passes through the point $P$ at a height of 1 m and horizontal distance of 5 m from $O$, as shown in the figure. If acceleration due to gravity is $g$ ms$^{-2}$, then the correct statement(s) is/are:

In a vacuum chamber, a particle of charge $1\ \mu$C and mass $1$ mg is projected with a velocity $(\hat\imath+2\hat\jmath)$ ms$^{-1}$ from the $XZ$ plane at time $t=0$ in an electric field of $1\hat\imath$ Vm$^{-1}$. At $t=0.2$ s, the electric field is switched off and a magnetic field of $6\hat\jmath$ T is switched on. The acceleration due to gravity is $-10\hat\jmath$ ms$^{-2}$. Correct option(s) is/are:

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.

A particle is rotating in a circular path and at any instant its motion can be described as $\theta=\dfrac{5t^4}{40}-\dfrac{t^3}3$. The angular acceleration of the particle after 10 seconds is ________ rad/s².

1 μC charge moving with velocity $\vec v=\left(\hat i-2\hat j+3\hat k\right)$ m/s in the region of magnetic field $\vec B=\left(2\hat i+3\hat j-5\hat k\right)$ T. The magnitude of force acting on it is $\sqrt\alpha\times10^{-6}$ N. The value of $\alpha$ is ________.

The time taken by a block of mass $m$ to slide down from the highest point to the lowest point on a rough inclined plane is 50 % more compared to the time taken by the same block on identical inclined smooth plane. Both inclined planes are at 45° with the horizontal. The coefficient of kinetic friction between the rough inclined surface and block is ________.

The two projectiles are projected with the same initial velocities at the 15° and 30° with respect to the horizontal. The ratio of their ranges is $1:x$. The value of $x$ is

A 1 kg block subjected to two simultaneous forces $\left(2\hat i+3\hat j+4\hat k\right)$ N and $\left(3\hat i-\hat j-2\hat k\right)$ N is moved a distance of 25 m along $\left(3\hat i-4\hat j\right)$ direction. The work done in this process is ________ J.

Two cars $A$ and $B$ are moving in the same direction along a straight line with speeds 100 km/h and 80 km/h, respectively such that car $A$ is moving ahead of car $B$. A person in car $B$ throws a stone with a speed $v$ so that it hits the car $A$ with a speed of 5 m/s. The value of $v$ is ________ km/h.

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 ________.

If $x$ and $y$ coordinates of a projectile as a function of time ($t$) are given as $24t$ and $43.6t-4.9t^2$, respectively, then the angle (in degrees) made by the projectile with horizontal when $t=2$ s is ________.

A gun mounted on the ground fires bullets in all directions with same speed. The farthest distance the bullets could reach is 6.4 m. The speed of the bullets from the gun is ________ m/s.
(take $g=10$ m/s²)

From 18 m height above the ground a ball is dropped from rest. The height above the ground at which the magnitude of velocity equal to the magnitude of acceleration (in the same set of units) due to gravity is ________ m.
(Take $g=10$ m/s² and neglect the air resistance)

The velocity ($v$) versus time ($t$) plot of a particle is shown in the figure, for a time interval of 40 s. The total distance travelled by the particle and the average velocity during this period are, respectively ________.

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 cylindrical vessel of 40 cm radius is completely filled with water and its capacity is 528 dm³ (dm : decimeter) The vessel is placed on a solid block of exactly same height as vessel. If a small hole is made at 70 cm below the top of water level, then horizontal range of water falling on the ground in the beginning is ________ cm.

The velocity at which 6 kg mass (shown in figure) strikes the ground when it is released from a height of 6 m above the ground is ________ m/s. Assume pulley is massless and string is light and inextensible. (Take $g=10$ m/s²)

A block takes $t$ time to slide down a plane inclined at 45° to the horizontal. If the surface is made smooth (frictionless), the block takes time $\dfrac t2$ to slide down the plane. The coefficient of friction between the block and the inclined plane is $\left(\dfrac\alpha{100}\right)$. The value of $\alpha$ is ________.

A gas balloon is going up with a constant velocity of 10 m/s. When this balloon reached a height of 75 m, a stone is dropped from it and balloon keeps moving up with the same velocity. The height of the balloon when the stone hits the ground is ________ m. (Take $g=10$ m/s²)