Projectile motion

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 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

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)

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.

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)

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 _____.

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?

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 _____.

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 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 $g=10\ \mathrm{ms^{-2}}$)

Given below are two statements. One is labelled as Assertion A and the other is labelled as Reason R.

Assertion A: Two identical balls $A$ and $B$ thrown with same velocity '$u$' at two different angles with horizontal attained the same range $R$. If $A$ and $B$ reached the maximum height $h_1$ and $h_2$ respectively, then $R=4\sqrt{h_1h_2}$

Reason R: Product of said heights.
$$h_1h_2=\left(\frac{u^2\sin^2\theta}{2g}\right)\cdot\left(\frac{u^2\cos^2\theta}{2g}\right)$$
Choose the correct answer :

A fighter jet is flying horizontally at a certain altitude with a speed of 200 m s$^{-1}$. When it passes directly overhead an anti-aircraft gun, a bullet is fired from the gun, at an angle $\theta$ with the horizontal, to hit the jet. If the bullet speed is 400 m/s, the value of $\theta$ will be ______°.

A projectile is launched at an angle '$\alpha$' with the horizontal with a velocity 20 ms$^{-1}$. After 10 s, its inclination with horizontal is '$\beta$'. The value of $\tan\beta$ will be : ($g=10$ ms$^{-2}$).

A person can throw a ball upto a maximum range of 100 m. How high above the ground he can throw the same ball?

Two projectiles $P_1$ and $P_2$ thrown with speed in the ratio $\sqrt3:\sqrt2$, attain the same height during their motion. If $P_2$ is thrown at an angle of 60° with the horizontal, the angle of projection of $P_1$ with horizontal will be :

A ball is projected from the ground with a speed 15 ms$^{-1}$ at an angle $\theta$ with horizontal so that its range and maximum height are equal, then '$\tan\theta$' will be equal to :