Projectiles

A ball is projected at an angle of $30°$ above with the horizontal from the top of a tower and strikes the ground in $5 \sec$ at angle of $45°$ with the horizontal. Find the height of the tower and the speed with which it was projected.

A particle is projected in x-y plane with y-axis along vertical, the point of projection being origin. The equation of projectile is $\text{y}=\sqrt{3}\text{x}-\frac{{\text{gx}}^2}{2 }$.  The angle of projectile is ---------------------- and initial velocity is -----------------------------------

A rifle with a muzzle velocity of $100 \mathrm{m} {\mathrm{s}}^{-1}$ shoots a bullet at a small target $30 \mathrm{m}$ away in the same horizontal line. How high above the target must the gun be aimed so that the bullet will hit the target?

The maximum range of a gun of horizontal terrain is $16 \mathrm{km}$. If $g=10\mathrm{m}{\mathrm{s}}^{-2},$ then, muzzle velocity of a shell must be

From a $10 \mathrm{m}$ high building, a stone $A$ is dropped and simultaneously another identical stone $B$ is thrown horizontally with an initial speed of  $5\mathrm{m} {\mathrm{s}}^{-1}.$ Which one of the following statements is true?

An object $A$ is kept fixed at the point  $x=3 m$ and  $y=1.25 m$ on a plank $P$ raised above the ground. At time  $t=0$  the plank starts moving along the  $+x$  direction with an acceleration  $1.5m{s}^{-2}.$  At the same instant a stone is projected from the origin with a velocity $u$ as shown. A stationary person on the ground observes the stone hitting the object during its downward motion at an angle of  $45°$  to the horizontal. All the motions are in the  $x-y$  plane. Find $u$ and the time after which the stone hits the object. Take  $g=10m{s}^{-2}.$          

A large heavy box is sliding without friction down a smooth plane of inclination  $\theta$ . From a point $P$ on the bottom of the box, a particle is projected inside the box. The initial speed of the particle with respect to the box is $u$ and the direction of projection makes an angle $\alpha$ with the bottom as shown in the figure:

If the horizontal displacement of the particle as seen by an observer on the ground is zero, find the speed of the box with respect to the ground at the instant when the particle was projected.

A large heavy box is sliding without friction down a smooth plane of inclination $\text{θ}$. From a point $\mathrm{P}$ on the bottom of the box, a particle is projected inside the box. The initial speed of the particle with respect to the box is $u$ and the direction of projection makes an angle $\alpha$ with the bottom as shown in the figure:



Find the distance along the bottom of the box between the point of projection $\mathrm{P}$ and the point $\mathrm{Q}$ where the particle lands. [Assume that the particle does not hit any other surface of the box. Neglect air resistance]

Two guns situated on the top of a hill of height $10 m$ fire one shot each with the same speed  $5\sqrt{3}m{s}^{-1}$ at some interval of time. One gun fires horizontally and other fires upwards at an angle of  $60°$  with the horizontal. The shots collide in air at point $P$. Find the time interval between the firings.

A projectile can have the same range $R$ for two angles of projection. If $t_1$ and $t_2$ be the times of flights in the two cases, then the product of the two time of flights is proportional to –

A projectile can have the same range ‘R’ for two angles of projection. If ‘t1’ and ‘t₂’ be the times of flight in the two cases, then the product of the two times of flight is proportional to 

A boy playing on the roof of a 10 m high building throws a ball with a speed of  $10m{s}^{-1 }$ at an angle of  $30°$ with the horizontal. How far from the throwing point will the ball be at the height of 10 m from the ground? [Take g = 10 ms^-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 $g^{\text{'}}=\frac{g}{0.81}$, then the new range is $d^{\text{'}}=nd.$ The value of $n$ is

The angular elevation of an enemy's position on a hill $sm$ above the gun position. Show that in order to shell it, the velocity of the projectile must not be less than $\sqrt{gs\left(1+cosec\beta \right)}$.

At an angle $\theta$ with horizontal, a particle is projected from surface of earth with velocity $V_0$ to attain the maximum height h:

A particle projected at an angle $\theta$ with the horizontal from the centre of the floor of a cylindrical room of radius $r$ returns to the point of projection after three elastic collisions with the walls and the ceiling. If the particle remains in air for time $T$, find the speed of projection.

In a tank filled with water, a small hole is made on the side. The emerging water jet falls to a distance of $0.8 \mathrm{m}$ from the base as shown $y$ is distance of hole from above) :-

A particle is projected on a frictionless inclined plane of inclination $37°$ at an angle of projection $45°$ from the inclined plane as shown in the figure. If after the first collision from the plane, the particle returns to its point of projection, then what is the value of the reciprocal of the coefficient of restitution between the particle and the plane?