The figure was extracted from an old computer game, called Bang! Bang!
In the game, two competitors control cannons A and B, firing bullets alternately with the objective of hitting the opponent's cannon; to do this, they assign estimated values for the magnitude of the initial velocity of firing ($\left|\overrightarrow{v_0}\right|$) and for the firing angle ($\theta$).
At a certain moment in a match, competitor B must fire; he knows that the bullet fired previously, $\theta = 53^{\circ}$, passed tangentially through point $\boldsymbol{P}$.
In the game, $|\vec{g}|$ equals $10 \mathrm{~m/s}^2$. Consider $\sin 53^{\circ} = 0.8$, $\cos 53^{\circ} = 0.6$ and negligible action of dissipative forces.
Based on the given distances and maintaining the last firing angle, what should be, approximately, the smallest value of $\left|\overrightarrow{v_0}\right|$ that would allow the shot fired by cannon $\mathbf{B}$ to hit cannon $\mathbf{A}$?
(A) $30 \mathrm{~m/s}$.
(B) $35 \mathrm{~m/s}$.
(C) $40 \mathrm{~m/s}$.
(D) $45 \mathrm{~m/s}$.
(E) $50 \mathrm{~m/s}$.

On a day of intense heat, two friends are playing with water from a hose. One of them wants to know how high the water jet reaches from the water outlet when the hose is positioned completely in the vertical direction. The other friend then proposes the following experiment: they position the hose water outlet in the horizontal direction, 1 m in height relative to the ground, and then measure the horizontal distance between the hose and the location where the water hits the ground. The measurement of this distance was 3 m, and from this they calculated the vertical reach of the water jet. Consider the acceleration due to gravity of $10 \mathrm{~m~s}^{-2}$.
The result they obtained was
(A) $1.50 \mathrm{~m}$.
(B) $2.25 \mathrm{~m}$.
(C) $4.00 \mathrm{~m}$.
(D) $4.50 \mathrm{~m}$.
(E) $5.00 \mathrm{~m}$.

A projectile is thrown from a point O on the ground at an angle $45^\circ$ from the vertical and with a speed $5\sqrt{2}$ m/s. The projectile at the highest point of its trajectory splits into two equal parts. One part falls vertically down to the ground, 0.5 s after the splitting. The other part, $t$ seconds after the splitting, falls to the ground at a distance $x$ meters from the point O. The acceleration due to gravity $g = 10$ m/s$^2$.
The value of $t$ is ____.

A projectile is thrown from a point O on the ground at an angle $45^\circ$ from the vertical and with a speed $5\sqrt{2}$ m/s. The projectile at the highest point of its trajectory splits into two equal parts. One part falls vertically down to the ground, 0.5 s after the splitting. The other part, $t$ seconds after the splitting, falls to the ground at a distance $x$ meters from the point O. The acceleration due to gravity $g = 10$ m/s$^2$.
The value of $x$ is ____.

A boy playing on the roof of a 10 m high building throws a ball with a speed of $10 \mathrm{~m/s}$ at an angle of $30^{\circ}$ with the horizontal. How far from the throwing point will the ball be at the height of 10 m from the ground?
$$\left[\mathrm{g} = 10 \mathrm{~m/s}^{2}, \sin 30^{\circ} = \frac{1}{2}, \cos 30^{\circ} = \frac{\sqrt{3}}{2}\right]$$
(1) 5.20 m
(2) 4.33 m
(3) 2.60 m
(4) 8.66 m

Two stones are projected from the top of a cliff $h$ metres high, with the same speed $u$, so as to hit the ground at the same spot. If one of the stones is projected at an angle $\theta$ to the horizontal then the $\theta$ equals
(1) $u\sqrt{\frac{2}{gh}}$
(2) $\sqrt{\frac{2u}{gh}}$
(3) $2g\sqrt{\frac{u}{h}}$
(4) $2h\sqrt{\frac{u}{g}}$

A ball is thrown from a point with a speed $v _ { 0 }$ at an angle of projection $\theta$. From the same point and at the same instant person starts running with a constant speed $v _ { 0 } / 2$ to catch the ball. Will the person be able to catch the ball? If yes, what should be the angle of projection?
(1) yes, $60 ^ { \circ }$
(2) yes, $30 ^ { \circ }$
(3) no
(4) yes, $45 ^ { \circ }$

A projectile can have the same range $R$ for two angles of projection. If $T _ { 1 }$ and $T _ { 2 }$ be the time of flights in the two cases, then the product of the two time of flights is directly proportional to
(1) $1 / R ^ { 2 }$
(2) $1 / R$
(3) R
(4) $R ^ { 2 }$

A projectile can have the same range R for two angles of projection. If $\mathrm{t}_1$ and $\mathrm{t}_2$ be the times of flights in the two cases, then the product of the two time of flights is proportional to
(1) $R^2$
(2) $1/R^2$
(3) $1/R$
(4) R

A particle is projected from a point O with velocity $u$ at an angle of $60^\circ$ with the horizontal. When it is moving in a direction at right angles to its direction at $O$, its velocity then is given by
(1) $\frac{u}{3}$
(2) $\frac{u}{2}$
(3) $\frac{2u}{3}$
(4) $\frac{u}{\sqrt{3}}$

A boy can throw a stone up to a maximum height of 10 m. The maximum horizontal distance that the boy can throw the same stone up to will be
(1) $20\sqrt{2}$ m
(2) 10 m
(3) $10\sqrt{2}$ m
(4) 20 m

A projectile moving vertically upwards with a velocity of $200 \mathrm{~ms}^{-1}$ breaks into two equal parts at a height of 490 m. One part starts moving vertically upwards with a velocity of $400 \mathrm{~ms}^{-1}$. How much time it will take, after the break up with the other part to hit the ground?
(1) $2\sqrt{10} \mathrm{~s}$
(2) 5 s
(3) 10 s
(4) $\sqrt{10} \mathrm{~s}$

The maximum range of a bullet fired from a toy pistol mounted on a car at rest is $R _ { 0 } = 40 \mathrm {~m}$. What will be the acute angle of inclination of the pistol for maximum range when the car is moving in the direction of firing with uniform velocity $\mathrm { v } = 20 \mathrm {~m} / \mathrm { s }$ on a horizontal surface? $\left( \mathrm { g } = 10 \mathrm {~m} / \mathrm { s } ^ { 2 } \right)$
(1) $30 ^ { \circ }$
(2) $60 ^ { \circ }$
(3) $75 ^ { \circ }$
(4) $45 ^ { \circ }$

A ball projected from ground at an angle of $45^{\circ}$ just clears a wall in front. If point of projection is 4 m from the foot of wall and ball strikes the ground at a distance of 6 m on the other side of the wall, the height of the wall is:
(1) 4.4 m
(2) 2.4 m
(3) 3.6 m
(4) 1.6 m

A tennis ball (treated as hollow spherical shell) starting from O rolls down a hill. At point A the ball becomes air borne leaving at an angle of $30^{\circ}$ with the horizontal. The ball strikes the ground at B. What is the value of the distance AB? (Moment of inertia of a spherical shell of mass $m$ and radius $R$ about its diameter $= \frac{2}{3}mR^2$)
(1) 1.87 m
(2) 2.08 m
(3) 1.57 m
(4) 1.77 m

The initial speed of a bullet fired from a rifle is $630 \mathrm{~m} / \mathrm{s}$. The rifle is fired at the centre of a target 700 m away at the same level as the target. How far above the centre of the target should the rifle be aimed so that the bullet hits the centre of the target? (Take $g = 10 \mathrm{~m/s}^2$)
(1) 1.0 m
(2) 4.2 m
(3) 6.1 m
(4) 9.8 m

An aeroplane flying at a constant speed, parallel to the horizontal ground, $\sqrt { 3 } \mathrm {~km}$ above it is observed at an elevation of $60 ^ { \circ }$ from a point on the ground. If after five seconds, its elevation from the same point is $30 ^ { \circ }$, then the speed (in $\mathrm { km } / \mathrm { hr }$ ) of the aeroplane is
(1) 720
(2) 1500
(3) 750
(4) 1440

An aeroplane flying at a constant speed, parallel to the horizontal ground, $\sqrt { 3 } \mathrm {~km}$ above it, is observed at an elevation of $60 ^ { \circ }$ from a point on the ground. If, after five seconds, its elevation from the same point, is $30 ^ { \circ }$, then the speed (in $\mathrm { km } / \mathrm { hr }$ ) of the aeroplane is
(1) 1500
(2) 750
(3) 720
(4) 1440

The trajectory of a projectile near the surface of the earth is given as $y = 2x - 9x^2$. If it were launched at an angle $\theta_0$ with speed $v_0$ then $g = 10 \text{ m s}^{-2}$:
(1) $\theta_0 = \cos^{-1}\frac{1}{\sqrt{5}}$ and $v_0 = \frac{5}{3} \text{ ms}^{-1}$
(2) $\theta_0 = \cos^{-1}\frac{2}{\sqrt{5}}$ and $v_0 = \frac{3}{5} \text{ ms}^{-1}$
(3) $\theta_0 = \sin^{-1}\frac{1}{\sqrt{5}}$ and $v_0 = \frac{5}{3} \text{ ms}^{-1}$
(4) $\theta_0 = \sin^{-1}\frac{2}{\sqrt{5}}$ and $v_0 = \frac{3}{5} \text{ ms}^{-1}$

Two particles are projected from the same point with the same speed $u$ such that they have the same range $R$, but different maximum heights, $\mathrm { h } _ { 1 }$ and $\mathrm { h } _ { 2 }$. Which of the following is correct?
(1) $R ^ { 2 } = h _ { 1 } h _ { 2 }$
(2) $R ^ { 2 } = 4 h _ { 1 } h _ { 2 }$
(3) $R ^ { 2 } = 2 h _ { 1 } h _ { 2 }$
(4) $R ^ { 2 } = 16 h _ { 1 } h _ { 2 }$

A body is projected at $t = 0$ with a velocity $10 \mathrm {~ms} ^ { - 1 }$ at an angle of $60 ^ { \circ }$ with the horizontal. The radius of curvature of its trajectory at $t = 1 \mathrm {~s}$ is $R$. Neglecting air resistance and taking acceleration due to gravity $\mathrm { g } = 10 \mathrm {~ms} ^ { - 2 }$, the value of $R$ is:
(1) 10.3 m
(2) 2.8 m
(3) 2.5 m
(4) 5.1 m

A shell is fired from a fixed artillery gun with an initial speed $u$ such that it hits the target on the ground at a distance $R$ from it. If $t_1$ and $t_2$ are the values of the time taken by it to hit the target in two possible ways, the product $t_1 t_2$ is:
(1) $R/2g$
(2) $R/g$
(3) $2R/g$
(4) $R/4g$

A plane is inclined at an angle $\alpha = 30 ^ { \circ }$ with respect to the horizontal. A particle is projected with a speed $\mathrm { u } = 2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, from the base of the plane, making an angle $\theta = 15 ^ { \circ }$ with respect to the plane as shown in the figure. The distance from the base, at which the particle hits the plane is close to: (Take $\mathrm { g } = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ )
(1) 20 cm
(2) 18 cm
(3) 14 cm
(4) 26 cm

A particle ($\mathrm { m } = 1 \mathrm {~kg}$) slides down a frictionless track (AOC) starting from rest at a point $A$ (height 2 m). After reaching $C$, the particle continues to move freely in air as a projectile. When it reaching its highest point P (height 1 m), the kinetic energy of the particle (in J) is: (Figure drawn is schematic and not to scale; take $g = 10 \mathrm {~ms} ^ { - 2 }$) $\_\_\_\_$.

The ranges and heights for two projectiles projected with the same initial velocity at angles $42 ^ { \circ }$ and $48 ^ { \circ }$ with the horizontal are $R _ { 1 } , \quad R _ { 2 }$ and $H _ { 1 } , \quad H _ { 2 }$ respectively. Choose the correct option:
(1) $R _ { 1 } = R _ { 2 }$ and $H _ { 1 } = H _ { 2 }$
(2) $R _ { 1 } = R _ { 2 }$ and $H _ { 1 } < H _ { 2 }$
(3) $R _ { 1 } > R _ { 2 }$ and $H _ { 1 } = H _ { 2 }$
(4) $R _ { 1 } < R _ { 2 }$ and $H _ { 1 } < H _ { 2 }$