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

If the angle of elevation of a cloud from a point $P$ which is $25 m$ above a lake be $30 ^ { \circ }$ and the angle of depression of reflection of the cloud in the lake from $P$ be $60 ^ { \circ }$, then the height of the cloud (in meters) from the surface of the lake is :
(1) 50
(2) 60
(3) 45
(4) 42

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 $H _ { 1 } < H _ { 2 }$

The trajectory of a projectile in a vertical plane is $y = \alpha x - \beta x ^ { 2 }$, where $\alpha$ and $\beta$ are constants and $x \& y$ are respectively the horizontal and vertical distances of the projectile from the point of projection. The angle of projection $\theta$ and the maximum height attained $H$ are respectively given by
(1) $\tan ^ { - 1 } \alpha , \frac { 4 \alpha ^ { 2 } } { \beta }$
(2) $\tan ^ { - 1 } \left( \frac { \beta } { \alpha } \right) , \frac { \alpha ^ { 2 } } { \beta }$
(3) $\tan ^ { - 1 } \beta , \frac { \alpha ^ { 2 } } { 2 \beta }$
(4) $\tan ^ { - 1 } \alpha , \frac { \alpha ^ { 2 } } { 4 \beta }$

A bomb is dropped by a fighter plane flying horizontally. To an observer sitting in the plane, the trajectory of the bomb is a:
(1) straight line vertically down the plane
(2) parabola in a direction opposite to the motion of plane
(3) parabola in the direction of motion of plane
(4) hyperbola

A swimmer can swim with velocity of $12 \mathrm {~km} / \mathrm { h }$ in still water. Water flowing in a river has velocity $6 \mathrm {~km} / \mathrm { h }$. The direction with respect to the direction of flow of river water he should swim in order to reach the point on the other bank just opposite to his starting point is $\_\_\_\_$. (Round off to the Nearest Integer) (find the angle in degree)

A pole stands vertically inside a triangular park $ABC$. Let the angle of elevation of the top of the pole from each corner of the park be $\frac { \pi } { 3 }$. If the radius of the circumcircle of $\triangle ABC$ is 2 , then the height of the pole is equal to:
(1) $\frac { 2 \sqrt { 3 } } { 3 }$
(2) $2 \sqrt { 3 }$
(3) $\sqrt { 3 }$
(4) $\frac { 1 } { \sqrt { 3 } }$

Two projectiles are thrown with same initial velocity making an angle of $45^{\circ}$ and $30^{\circ}$ with the horizontal respectively. The ratio of their respective ranges will be
(1) $1 : \sqrt { 2 }$
(2) $\sqrt { 2 } : 1$
(3) $2 : \sqrt { 3 }$
(4) $\sqrt { 3 } : 2$

A projectile is projected with velocity of $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ 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 $\mathrm { g } = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ ]
(1) $\frac { 1 } { 2 } \sin ^ { - 1 } \left( \frac { 5 t ^ { 2 } } { 4 R } \right)$
(2) $\frac { 1 } { 2 } \sin ^ { - 1 } \left( \frac { 4 R } { 5 t ^ { 2 } } \right)$
(3) $\tan ^ { - 1 } \left( \frac { 4 t ^ { 2 } } { 5 R } \right)$
(4) $\cot ^ { - 1 } \left( \frac { R } { 20 t ^ { 2 } } \right)$

A ball is projected from the ground with a speed $15 \mathrm{~m}\mathrm{~s}^{-1}$ at an angle $\theta$ with horizontal so that its range and maximum height are equal, then $\tan\theta$ will be equal to
(1) $\frac{1}{4}$
(2) $\frac{1}{2}$
(3) 2
(4) 4

A person can throw a ball upto a maximum range of 100 m. How high above the ground he can throw the same ball?
(1) 25 m
(2) 50 m
(3) 100 m
(4) 200 m

An object is projected in the air with initial velocity $u$ at an angle $\theta$. The projectile motion is such that the horizontal range $R$, is maximum. Another object is projected in the air with a horizontal range half of the range of first object. The initial velocity remains same in both the case. The value of the angle of projection, at which the second object is projected, will be $\_\_\_\_$ degree.

Given below are two statements: one is labelled as Assertion $A$ and the other is labelled as Reason $R$. Assertion A: When a body is projected at an angle $45^{\circ}$, its range is maximum. Reason R: For maximum range, the value of $\sin 2\theta$ should be equal to one. In the light of the above statements, choose the correct answer from the options given below:
(1) $A$ is false but $R$ is true
(2) $A$ is true but $R$ is false
(3) Both $A$ and $R$ are correct and $R$ is the correct explanation of $A$
(4) Both $A$ and $R$ are correct but $R$ is NOT the correct explanation of $A$

A projectile fired at $30^\circ$ to the ground is observed to be at same height at time 3 s and 5 s after projection, during its flight. The speed of projection of the projectile is $\_\_\_\_$ $\mathrm{m}\mathrm{~s}^{-1}$. (Given $g = 10$ m s$^{-2}$)

A particle of mass $m$ projected with a velocity $u$ making an angle of $30 ^ { \circ }$ with the horizontal. The magnitude of angular momentum of the projectile about the point of projection when the particle is at its maximum height $h$ is :
(1) $\frac { \sqrt { 3 } } { 16 } \frac { m u ^ { 3 } } { g }$
(2) $\frac { \sqrt { 3 } } { 2 } \frac { m u ^ { 2 } } { g }$
(3) $\frac { m u ^ { 3 } } { \sqrt { 2 } g }$
(4) zero

A body of mass $m$ is projected with a speed $u$ making an angle of $45^\circ$ with the ground. The angular momentum of the body about the point of projection, at the highest point is expressed as $\dfrac{\sqrt{Z}\, m u^3}{X g}$. The value of $X$ is $\_\_\_\_$.

Q2. The angle of projection for a projectile to have same horizontal range and maximum height is :
(1) $\tan ^ { - 1 } ( 4 )$
(2) $\tan ^ { - 1 } \left( \frac { 1 } { 4 } \right)$
(3) $\tan ^ { - 1 } \left( \frac { 1 } { 2 } \right)$
(4) $\tan ^ { - 1 } ( 2 )$

Q21. The maximum height reached by a projectile is 64 m . If the initial velocity is halved, the new maximum height of the projectile is $\_\_\_\_$ m.

Two projectiles are fired with same initial speed from same point on ground at angles of $(45^\circ - \alpha)$ and $(45^\circ + \alpha)$, respectively, with the horizontal direction. The ratio of their maximum heights attained is:
(1) $\frac{1 - \tan\alpha}{1 + \tan\alpha}$
(2) $\frac{1 - \sin 2\alpha}{1 + \sin 2\alpha}$
(3) $\frac{1 + \sin 2\alpha}{1 - \sin 2\alpha}$
(4) $\frac{1 + \sin\alpha}{1 - \sin\alpha}$

The maximum speed of a boat in still water is $27\,\mathrm{km/h}$. Now this boat is moving downstream in a river flowing at $9\,\mathrm{km/h}$. A man in the boat throws a ball vertically upwards with speed of $10\,\mathrm{m/s}$. Range of the ball as observed by an observer at rest on the river bank, is \_\_\_\_ cm. (Take $g = 10\,\mathrm{m/s^2}$)

A projectile is projected at angle of projection $60 ^ { \circ }$ with speed $u$. When its velocity makes an angle $45 ^ { \circ }$ with horizontal its speed is $20 \mathrm {~m} / \mathrm { s }$. Find u ?
(A) $\mathbf { 1 0 } \sqrt { \mathbf { 2 } }$
(B) $20 \mathrm {~m} / \mathrm { s }$
(C) $20 \sqrt { 2 } \mathrm {~m} / \mathrm { s }$
(D) $40 \mathrm {~m} / \mathrm { s }$

On the same plane, two artillery batteries $A$ and $B$ are 7 kilometers apart, with $A$ directly east of $B$. During an exercise, $A$ fires a projectile west-northwest at angle $\theta$, and $B$ fires a projectile east-northwest at angle $\theta$, where $\theta$ is an acute angle. Both projectiles hit the same target $P$ 9 kilometers away. Then $A$ fires another projectile west-northwest at angle $\frac{\theta}{2}$, landing at point $Q$ 9 kilometers away. What is the distance $\overline{BQ}$ between artillery battery $B$ and landing point $Q$?
(1) 4 kilometers
(2) 4.5 kilometers
(3) 5 kilometers
(4) 5.5 kilometers
(5) 6 kilometers