Kinematics

Two identical bodies, projected with the same speed at two different angles cover the same horizontal range $R$. If the time of flight of these bodies are 5 s and 10 s, respectively, then the value of $R$ is ________ m. (Take $g=10$ m/s²)

Two masses of 3.4 kg and 2.5 kg are accelerated from an initial speed of 5 m/s and 12 m/s, respectively. The distances traversed by the masses in the 5th second are 104 m and 129 m, respectively. The ratio of their momenta after 10 s is $\dfrac x8$. The value of $x$ is ________.

A geostationary satellite above the equator is orbiting around the earth at a fixed distance $r_1$ from the center of the earth. A second satellite is orbiting in the equatorial plane in the opposite direction to the earth's rotation, at a distance $r_2$ from the center of the earth, such that $r_1 = 1.21\,r_2$. The time period of the second satellite as measured from the geostationary satellite is $\dfrac{24}{p}$ hours. The value of $p$ is _____.

A projectile of mass 200 g is launched in a viscous medium at an angle 60° with the horizontal, with an initial velocity of 270 m/s. It experiences a viscous drag force $\vec F = -c\vec v$ where the drag coefficient $c = 0.1$ kg/s and $\vec v$ is the instantaneous velocity of the projectile. The projectile hits a vertical wall after 2 s. Taking $e = 2.7$, the horizontal distance of the wall from the point of projection (in m) is _____.

A ball is thrown from the location $(x_0, y_0) = (0, 0)$ of a horizontal playground with an initial speed $v_0$ at an angle $\theta_0$ from the $+x$-direction. The ball is to be hit by a stone, which is thrown at the same time from the location $(x_1, y_1) = (L, 0)$. The stone is thrown at an angle $(180 - \theta_1)$ from the $+x$-direction with a suitable initial speed. For a fixed $v_0$, when $(\theta_0, \theta_1) = (45^{\circ}, 45^{\circ})$, the stone hits the ball after time $T_1$, and when $(\theta_0, \theta_1) = (60^{\circ}, 30^{\circ})$, it hits the ball after time $T_2$. In such a case, $(T_1/T_2)^2$ is _____.

The maximum speed in $\mu$m/s at which the $8^{\text{th}}$ bright fringe will move is _____.

A slide with a frictionless curved surface, which becomes horizontal at its lower end, is fixed on the terrace of a building of height $3h$ from the ground, as shown in the figure. A spherical ball of mass $m$ is released on the slide from rest at a height $h$ from the top of the terrace. The ball leaves the slide with a velocity $\vec u_0 = u_0\hat x$ and falls on the ground at a distance $d$ from the building making an angle $\theta$ with the horizontal. It bounces off with a velocity $\vec v$ and reaches a maximum height $h_1$. The acceleration due to gravity is $g$ and the coefficient of restitution of the ground is $1/\sqrt3$. Which of the following statement(s) is(are) correct?

A person of height 1.6 m is walking away from a lamp post of height 4 m along a straight path on the flat ground. The lamp post and the person are always perpendicular to the ground. If the speed of the person is $60\ \mathrm{cm\,s^{-1}}$, the speed of the tip of the person's shadow on the ground with respect to the person is _____ $\mathrm{cm\,s^{-1}}$.

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 thin circular coin of mass 5 gm and radius $4/3$ cm is initially in a horizontal $xy$-plane. The coin is tossed vertically up ($+z$ direction) by applying an impulse of $\sqrt{\dfrac\pi2}\times10^{-2}$ N-s at a distance $2/3$ cm from its center. The coin spins about its diameter and moves along the $+z$ direction. By the time the coin reaches back to its initial position, it completes $n$ rotations. The value of $n$ is _____.

[Given: The acceleration due to gravity $g = 10\ \mathrm{m\,s^{-2}}$]

When only $S_2$ is emitting sound and it is at $Q$, the frequency of sound measured by the detector in Hz is _____.

At time $t = 0$, a disk of radius 1 m starts to roll without slipping on a horizontal plane with an angular acceleration of $\alpha = \dfrac23\ \mathrm{rad\,s^{-2}}$. A small stone is stuck to the disk. At $t = 0$, it is at the contact point of the disk and the plane. Later, at time $t = \sqrt\pi$ s, the stone detaches itself and flies off tangentially from the disk. The maximum height (in m) reached by the stone measured from the plane is $\dfrac12 + \dfrac{x}{10}$. The value of $x$ is _____. [Take $g = 10\ \mathrm{m\,s^{-2}}$.]

A projectile is fired from horizontal ground with speed $v$ and projection angle $\theta$. When the acceleration due to gravity is $g$, the range of the projectile is $d$. If at the highest point in its trajectory, the projectile enters a different region where the effective acceleration due to gravity is $g' = \dfrac{g}{0.81}$, then the new range is $d' = nd$. The value of $n$ is _____.

List I describes four systems, each with two particles $A$ and $B$ in relative motion as shown in figures. List II gives possible magnitudes of their relative velocities (in $\mathrm{m\,s^{-1}}$) at time $t = \dfrac\pi3$ s.

Which one of the following options is correct?

An object and a concave mirror of focal length $f = 10$ cm both move along the principal axis of the mirror with constant speeds. The object moves with speed $V_0 = 15\ \mathrm{cm\,s^{-1}}$ towards the mirror with respect to a laboratory frame. The distance between the object and the mirror at a given moment is denoted by $u$. When $u = 30$ cm, the speed of the mirror $V_m$ is such that the image is instantaneously at rest with respect to the laboratory frame, and the object forms a real image. The magnitude of $V_m$ is _____ $\mathrm{cm\,s^{-1}}$.

A projectile is projected with velocity of 25 m/s at an angle $\theta$ with the horizontal. After $t$ seconds its inclination with horizontal becomes zero. If $R$ represents horizontal range of the projectile, the value of $\theta$ will be :

[use $g=10\ \mathrm{m/s^2}$]

A block of mass 10 kg starts sliding on a surface with an initial velocity of $9.8\ \mathrm{ms^{-1}}$. The coefficient of friction between the surface and block is 0.5. The distance covered by the block before coming to rest is :

[use $g=9.8\ \mathrm{ms^{-2}}$]

From the top of a tower, a ball is thrown vertically upward which reaches the ground in 6 s. A second ball thrown vertically downward from the same position with the same speed reaches the ground in 1.5 s. A third ball released, from the rest from the same location, will reach the ground in ______ s.

An object of mass 5 kg is thrown vertically upwards from the ground. The air resistance produces a constant retarding force of 10 N throughout the motion. The ratio of time of ascent to the time of descent will be equal to : [Use $g=10\ \mathrm{ms^{-2}}$].

A body is projected from the ground at an angle of 45° with the horizontal. Its velocity after 2s is $20\ \mathrm{ms^{-1}}$. The maximum height reached by the body during its motion is ______ m. (use $g=10\ \mathrm{ms^{-2}}$)