A projectile is thrown upward at an angle $60^{\circ}$ with the horizontal. The speed of the projectile is $20 \mathrm{~m} / \mathrm{s}$ when its direction of motion is $45^{\circ}$ with the horizontal. The initial speed of the projectile is $\_\_\_\_$ $\mathrm{m} / \mathrm{s}$.

A river of width 200 m is flowing from west to east with a speed of 18 km/h. A boat, moving with speed of 36 km/h in still water, is made to travel one-round trip (bank to bank of the river). Minimum time taken by the boat for this journey and also the displacement along the river bank are ______ and ______ respectively.

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 $g=10 \mathrm{~m} / \mathrm{s}^2$ )

At $t=0$, a body of mass 100 g starts moving under the influence of a force $(5 \hat{\mathrm{i}}+10 \hat{\mathrm{j}}) \mathrm{N} \cdot$ After 2 s its position is $(2 x \hat{\mathrm{i}}+5 y \hat{\mathrm{j}}) \mathrm{m}$. The ratio $x: y$ is $\_\_\_\_$ .

If $x$ and $y$ coordinates of a projectile as a function of time $(t)$ are given as $24 t$ and $43.6 t-4.9 t^2$, respectively, then the angle (in degrees) made by the projectile with horizontal when $t=2 \mathrm{~s}$ is $\_\_\_\_$ .

The two projectiles are projected with the same initial velocities at the $15^{\circ}$ and $30^{\circ}$ with respect to the horizontal. The ratio of their ranges is $1: x$. The value of $x$ is

The velocity of a particle is given as $\vec{v} = -x \hat{i} + 2y \hat{j} - z \hat{k}$ m/s. The magnitude of acceleration at point (1, 2, 4) is ________ m/s².

A helicopter flying horizontally with a speed of 360 km/h at an altitude of 2 km, drops an object at an instant. The object hits the ground at a point O, 20 s after it is dropped. Displacement of 'O' from the position of helicopter where the object was released is :

(use acceleration due to gravity g = 10 m/s² and neglect air resistance)

Two projectiles are fired from ground with same initial speeds from same point at angles $\left(45^{\circ}+\right.$ $\alpha)$ and $\left(45^{\circ}-\alpha\right)$ with horizontal direction. The ratio of their times of flights is

A particle is projected with velocity $u$ so that its horizontal range is three times the maximum height attained by it. The horizontal range of the projectile is given as $\frac{n u^2}{25 g}$, where value of $n$ is: (Given, ' $g$ ' is the acceleration due to gravity.)

The angle of projection of a particle is measured from the vertical axis as $\phi$ and the maximum height reached by the particle is $\mathrm{h}_{\mathrm{m}}$. Here $\mathrm{h}_{\mathrm{m}}$ as function of $\phi$ can be presented as

A river is flowing from west to east direction with speed of $9 \mathrm{~km} \mathrm{~h}^{-1}$. If a boat capable of moving at a maximum speed of $27 \mathrm{~km} \mathrm{~h}^{-1}$ in still water, crosses the river in half a minute, while moving with maximum speed at an angle of $150^{\circ}$ to direction of river flow, then the width of the river is :

The position vector of a moving body at any instant of time is given as $\overrightarrow{\mathrm{r}}=\left(5 \mathrm{t}^2 \hat{i}-5 \mathrm{t} \hat{j}\right) \mathrm{m}$. The magnitude and direction of velocity at $t=2 s$ is,

An object of mass ' m ' is projected from origin in a vertical xy plane at an angle $45^{\circ}$ with the $\mathrm{x}-$ axis with an initial velocity $\mathrm{v}_0$. The magnitude and direction of the angular momentum of the object with respect to origin, when it reaches at the maximum height, will be [ g is acceleration due to gravity]

A ball of mass 100 g is projected with velocity $20 \mathrm{~m} / \mathrm{s}$ at $60^{\circ}$ with horizontal. The decrease in kinetic energy of the ball during the motion from point of projection to highest point is

The angle of projection for a projectile to have same horizontal range and maximum height is :

The co-ordinates of a particle moving in $x$-$y$ plane are given by : $x=2+4 \mathrm{t}, y=3 \mathrm{t}+8 \mathrm{t}^2$.

The motion of the particle is :

Projectiles A and B are thrown at angles of $45^{\circ}$ and $60^{\circ}$ with vertical respectively from top of a $400 \mathrm{~m}$ high tower. If their ranges and times of flight are same, the ratio of their speeds of projection $v_A: v_B$ is :

[Take $g=10 \mathrm{~ms}^{-2}$]

Position of an ant ($\mathrm{S}$ in metres) moving in $\mathrm{Y}$-$\mathrm{Z}$ plane is given by $S=2 t^2 \hat{j}+5 \hat{k}$ (where $t$ is in second). The magnitude and direction of velocity of the ant at $\mathrm{t}=1 \mathrm{~s}$ will be :

A projectile is projected at $30^{\circ}$ from horizontal with initial velocity $40 \mathrm{~ms}^{-1}$. The velocity of the projectile at $\mathrm{t}=2 \mathrm{~s}$ from the start will be : (Given $g=10 \mathrm{~m} / \mathrm{s}^{2}$ )

Two projectiles are projected at $30^{\circ}$ and $60^{\circ}$ with the horizontal with the same speed. The ratio of the maximum height attained by the two projectiles respectively is:

The range of the projectile projected at an angle of 15$^\circ$ with horizontal is 50 m. If the projectile is projected with same velocity at an angle of 45$^\circ$ with horizontal, then its range will be

The trajectory of projectile, projected from the ground is given by $y=x-\frac{x^{2}}{20}$. Where $x$ and $y$ are measured in meter. The maximum height attained by the projectile will be.

Two projectiles A and B are thrown with initial velocities of $40 \mathrm{~m} / \mathrm{s}$ and $60 \mathrm{~m} / \mathrm{s}$ at angles $30^{\circ}$ and $60^{\circ}$ with the horizontal respectively. The ratio of their ranges respectively is $\left(g=10 \mathrm{~m} / \mathrm{s}^{2}\right)$

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 $45^{\circ}$, it's 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:

A child stands on the edge of the cliff $10 \mathrm{~m}$ above the ground and throws a stone horizontally with an initial speed of $5 \mathrm{~ms}^{-1}$. Neglecting the air resistance, the speed with which the stone hits the ground will be $\mathrm{ms}^{-1}$ (given, $g=10 \mathrm{~ms}^{-2}$ ).

The initial speed of a projectile fired from ground is $\mathrm{u}$. At the highest point during its motion, the speed of projectile is $\frac{\sqrt{3}}{2} u$. The time of flight of the projectile is :

Two objects are projected with same velocity 'u' however at different angles $\alpha$ and $\beta$ with the horizontal. If $\alpha+\beta=90^\circ$, the ratio of horizontal range of the first object to the 2nd object will be :

The maximum vertical height to which a man can throw a ball is 136 m. The maximum horizontal distance upto which he can throw the same ball is :

At time $t=0$ a particle starts travelling from a height $7 \hat{z} \mathrm{~cm}$ in a plane keeping z coordinate constant. At any instant of time it's position along the $\hat{x}$ and $\hat{y}$ directions are defined as $3 \mathrm{t}$ and $5 \mathrm{t}^{3}$ respectively. At t = 1s acceleration of the particle will be

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 :

Two projectiles thrown at $30^{\circ}$ and $45^{\circ}$ with the horizontal respectively, reach the maximum height in same time. The ratio of their initial velocities is :

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 :

At t = 0, truck, starting from rest, moves in the positive x-direction at uniform acceleration of 5 ms^$-$2. At t = 20 s, a ball is released from the top of the truck. The ball strikes the ground in 1 s after the release. The velocity of the ball, when it strikes the ground, will be :

(Given g = 10 ms^$-$2)

Two projectiles P₁ and P₂ thrown with speed in the ratio $\sqrt3$ : $\sqrt2$, attain the same height during their motion. If P₂ is thrown at an angle of 60$^\circ$ with the horizontal, the angle of projection of P₁ with horizontal will be :

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

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 girl standing on road holds her umbrella at 45$^\circ$ with the vertical to keep the rain away. If she starts running without umbrella with a speed of 15$\sqrt2$ kmh^$-$1, the rain drops hit her head vertically. The speed of rain drops with respect to the moving girl is :

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 h₁ and h₂ respectively, then $R = 4\sqrt {{h_1}{h_2}} $

Reason R : Product of said heights.

${h_1}{h_2} = \left( {{{{u^2}{{\sin }^2}\theta } \over {2g}}} \right)\,.\,\left( {{{{u^2}{{\cos }^2}\theta } \over {2g}}} \right)$

Choose the correct answer :

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 m/s²]