Motion in a Plane

A body projected at certain angle $\left(\neq 90^{\circ}\right)$ from the ground crosses a point in its path at a time of 2.3 s and from there it reaches the ground after a time of 5.7 s . The maximum heigh reached by the body is (Accelerat...

The vertical displacement ( $y$ in metre) of a projectile in term of its horizontal displacement ( $x$ in metre) is given by $y=\left(\sqrt{3} x-0.2 x^2\right)$. The time of flight of the projectile is
(Acceleration due to gravity $=10 \mat...

If a body projected with a velocity of $19.6 \mathrm{~ms}^{-1}$ reaches a maximum height of 9.8 m , then the range of the projectile is
(Neglect air resistance)

A ball projected at an angle of $45^{\circ}$ with the horizontal crosses two points at equal heights separated by a distance at times 2 s and 8 s respectively. The horizontal distance between the two points is
(Acceleration due to gravity $...

A helicopter flying horizontally with a velocity of $288 \mathrm{~km} / \mathrm{h}$ drops a bomb. If the line joining the point of dropping the bomb and the point where bomb hits the ground makes an angle $45^{\circ}$ with the horizontal, t...

Two bodies are projected from the same point with the same initial velocity ' $u$ ' making angles ' $\theta^{\prime}$ and $\left(90^{\circ}-\theta\right)$ with the horizontal in opposite directions. The horizontal distance between their pos...

A body $P$ is projected at an angle of $30^{\circ}$ with the horizontal and another body $Q$ is projected at an angle of $30^{\circ}$ with the vertical. If the ratio of the horizontal ranges of the bodies $P$ and $Q$ is $1: 2$, then the rat...

A cannon placed on a cliff at a height of 375 m fires a cannon ball with a velocity of $100 \mathrm{~ms}^{-1}$ at an angle of $30^{\circ}$ above the horizontal. The horizontal distance between the cannon and the target is (Acceleration due ...

The maximum horizontal range of a ball projected from the ground is 32 m . If the ball is thrown with the same speed horizontally from the top of a tower of . height 25 m , the maximum horizontal distance covered by the ball is(Acceleration...

A body is projected from the ground at an angle of $\tan ^{-1}(\sqrt{7})$ with the horizontal. At half of the maximum height, the speed of the body is $n$ times the speed of projection. The value of $n$ is

From a height of $h$ above the ground, a ball is projected up at an angle $30^{\circ}$ with the horizontal. If the ball strikes the ground with a speed of 1.25 times its initial speed of $40 \mathrm{~ms}^{-1}$, the value of $h$ is (accelera...

The velocity of a particle having magnitude of $10 \mathrm{~ms}^{-1}$ in the direction of $60^{\circ}$ with positive $X$-axis is

A player can throw a ball to a maximum horizontal distance of 80 m . If he throws the ball vertically with the same velocity, then the maximum height reached by the ball is

A projectile is given an initial velocity of $\hat{\mathbf{i}}+2 \hat{\mathbf{j}} \mathrm{~ms}^{-1}$. The cartesian equation of its path is ( $x$ and $y$ are in metres and $g=10 \mathrm{~ms}^{-1}$ )

A stone projected from the ground with a velocity $50 \mathrm{~ms}^{-1}$ at an angle of $30^{\circ}$ with the horizontal crosses a wall after a time of 3 s . Then the horizontal distance beyond the wall that the stone strikes the ground is ...

A bomb is dropped on an enemy post on the ground by an aeroplane flying horizontally with a velocity of $60 \mathrm{kmh}^{-1}$ at a height of 490 m . At the time of dropping the bomb, the horizontal distance of the aeroplane from the enemy ...

A person walks in such a way that he covers equal distance in each step. The person takes 2 steps forwards towards east, then takes a right turn and
walks 4 steps towards south, then takes a right turn and walks 6 steps towards west and the...

For a projectile, if $\alpha$ is the angle of projection, $R$ is the range, $h$ is the maximum height, $t$ is the time of flight then

Statement I An object subjected to velocities $\mathbf{v}_1$ and $\mathbf{v}_2$ has a resultant velocity with magnitude $|\mathbf{v}|=\left|\mathbf{v}_1\right|+\left|\mathbf{v}_2\right|$.
Statement II The magnitude of displacement is either...

The surface of a hill inclined at an angle $30^{\circ}$ to the horizontal. A stone is thrown from the summit of the hill (point $A$ ) at an initial speed $10 \mathrm{~m} / \mathrm{s}$ at angle $60^{\circ}$ to the vertical. If the stone stri...

Particle $A$ (which was located at the origin at time $t=0$ ) is moving along the $X$-axis with a constant speed of $1 \mathrm{~m} / \mathrm{s}$. Location of particle $B$ which is moving along the $Y$-axis is given by $y=c t^2$, where $c=1 ...

A particle is moving in $X Y$-plane as $\mathbf{x}=\left(4 t+t^2\right) \hat{\mathbf{i}}$, $\mathbf{y}=\left(2 t+\frac{t^2}{2}\right) \hat{\mathbf{j}}$, where $\mathbf{x}$ and $\mathbf{y}$ are displacements measured along $X$ and $Y$-axes r...

A man walking along a straight line with a velocity 6 $\mathrm{km} / \mathrm{h}$ encounters rain falling vertically down with a velocity $6 \sqrt{3} \mathrm{~km} / \mathrm{h}$. At what angle the man should hold his umbrella, so that he can ...

A bullet is fired at time $t=0$ with velocity $20 \mathrm{~m} / \mathrm{s}$ and at an initial angle of $30^{\circ}$ with the horizontal. The angle between the displacement vector and the horizontal after time 0.1 s is (assume $g=10 \mathrm{...

Two cars, at a certain instant, are 50 km apart on a line running from south to north. The one farther north is moving west at $25 \mathrm{~km} / \mathrm{h}$. The other is moving towards north at $25 \mathrm{~km} / \mathrm{h}$. How long do ...

A particle initially at origin starts moving in $X Y$ - plane has velocity component $\mathbf{v}=(6+2 t) \hat{\mathbf{i}}+(4+2 \sqrt{3 t}) \hat{\mathbf{j}} \mathrm{m} / \mathrm{s}$. Acceleration of the particle in $\mathrm{m} / \mathrm{s}^...

A projectile is launched with an initial speed of $40 \mathrm{~m} / \mathrm{s}$ at an angle $30^{\circ}$ above the ground. The projectile lands on a hillside 2.0 s later. The net displacement from where the projectile lands on hillside 2.0 ...

A projectile is given an initial velocity of $(3 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}) \mathrm{m} / \mathrm{s}$ where $\hat{\mathbf{i}}$ is along the ground and $\hat{\mathbf{j}}$ is along the vertical. Assuming $g=10 \mathrm{~m} / \mathrm{s...

A small object slides down with initial velocity equal to zero from the top of a smooth hill of height $H$. The other end of the hill is horizontal and is at height $H / 2$ as shown in the figure. The horizontal distance covered by the obje...

A ball is projected with a velocity $5 \mathrm{~m} / \mathrm{s}$, so that its horizontal range is twice the greatest height attained. The value of range is

A point moves in the $x y$-plane according to the following equation, $x=a \sin \omega t, y=a(1-\cos \omega t)$, where $a$ and $\omega$ are positive constants. Find the angle between the point's velocity and acceleration vectors.

A person moves 30 m North and then 20 m towards East and finally $30 \sqrt{2} \mathrm{~m}$ in South-West direction. The displacement of the person from the origin will be

A river 200 m wide is flowing at a rate of $3.0 \mathrm{~m} / \mathrm{s}$. A boat is sailing at a velocity of $15 \mathrm{~m} / \mathrm{s}$ with respect to the water in a direction perpendicular to the river. How far from the point directly...

A particle aimed at a target, projected with an angle $15^{\circ}$ with the horizontal is short of the target by 10 m . If projected with an angle of $45^{\circ}$ is away from the target by 10 m , then the angle of projection to hit the tar...

A particle of mass $m=1 \mathrm{~kg}$ moves in the $x y$-plane. The force on it at time $t$ is $F(t)=[2 \sin (\alpha t) \hat{\mathbf{i}}+3 \cos (\alpha t) \hat{\mathbf{j}}] \mathrm{N}$, where $\alpha=1 \mathrm{~s}^{-1}$. At time $t=0$, the ...

A projectile is launched from point $A$ of the given landscape with a water body as shown in the diagram. The launching angle is $15^{\circ}$. From the following, identify the right initial velocity of the projectile with which it will fall...

Initial velocity with which a body is projected is $10 \mathrm{~m} / \mathrm{s}$ from the base of an inclined plane as shown in the given figure. If the angle of projection is $60^{\circ}$ with the horizontal, then the range $R$ is [take, $...

A projectile is fired at an angle of $45^{\circ}$ with the horizontal. Elevation angle of the projectile at its highest point as seen from the point of projection is

Two cars $A$ and $B$ are moving with speeds $v_A=120 \mathrm{km} / \mathrm{h}$ and $v_B=50 \mathrm{~km} / \mathrm{h}$ respectively in the directions as indicated by the arrow in the figure below. What is the relative speed of the car $B$ wi...