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Two bodies are projected from ground with same speeds \( 40 \, \text{m s}^{-1} \) at two different angles with respect to horizontal. The bodies were found to have same range. If one of the body was projected at an angle of \( 60^\circ \), with horizontal then sum of the maximum heights, attained by the two projectiles, is ______ m.
(Given \( g = 10 \, \text{m s}^{-2} \))

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

A ball is spun with angular acceleration \( \alpha = 6t^2 - 2t \) where \( t \) is in second and \( \alpha \) is in \( \text{rad s}^{-2} \). At \( t = 0 \), the ball has angular velocity of \( 10 \, \text{rad s}^{-1} \) and angular position of \( 4 \, \text{rad} \). The most appropriate expression for the angular position of the ball is:

A given object takes \( n \) times the time to slide down \( 45^\circ \) rough inclined plane as it takes the time to slide down an identical perfectly smooth \( 45^\circ \) inclined plane. The coefficient of kinetic friction between the object and the surface of inclined plane is:

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

A body of mass \( 10 \, \text{kg} \) is projected at an angle of \( 45^\circ \) with the horizontal. The trajectory of the body is observed to pass through a point \( (20, 10) \). If \( T \) is the time of flight, then its momentum vector, at time \( t = \dfrac{T}{\sqrt{2}} \), is __________ [Take \( g = 10 \, \text{m}/\text{s}^{2} \)]

A fly wheel is accelerated uniformly from rest and rotates through \( 5 \,\text{rad} \) in the first second. The angle rotated by the fly wheel in the next second, will be:

For a body projected at an angle with the horizontal from the ground, choose the correct statement.

A particle of mass \( M \) originally at rest is subjected to a force whose direction is constant but magnitude varies with time according to the relation \( F = F_{0} \left[ 1 - \left( \dfrac{t -T}{T} \right)^{2} \right] \). Where \( F_{0} \) and \( T \) are constants. The force acts only for the time interval \( 2T \). The velocity \( v \) of the particle after time \( 2T \) is:

A body is projected from the ground at an angle of \( 45^\circ \) with the horizontal. Its velocity after \( 2 \, \text{s} \) is \( 20 \, \text{ms}^{-1} \). The maximum height reached by the body during its motion is _______ \( \text{m} \). (use \( g = 10 \, \text{ms}^{-2} \) )

As shown in the figure, a particle is moving with constant speed \( \pi \, \text{m/s} \). Considering its motion from \( A \) to \( B \), the magnitude of the average velocity is:

Initial velocity \( \vec{u} \) and acceleration \( \vec{a} \) of a particle are given as \( \vec{u}=(2\hat{i}+5\hat{j}) \, \text{m}/\text{s} \; ; \; \vec{a}=5\hat{i} \, \text{m}/\text{s}^2 \)
Given below are two statements: One is labelled as Assertion (A) and other is labelled as Reason (R).
Assertion (A) : Trajectory of particle is parabolic.
Reason (R) : If motion in \( 2 \)-Dimension is uniform and uniformly accelerated along mutually perpendicular directions, then path followed is parabolic.
In the light of above statements, choose the correct answer from the options given below.

For a train engine moving with speed of \( 20 \, \text{m s}^{-1} \), the driver must apply brakes at a distance of \( 500 \, \text{m} \) before the station for the train to come to rest at the station. If the brakes were applied at half of this distance, the train engine would cross the station with speed \( \sqrt{x} \, \text{m s}^{-1} \). The value of \( x \) is _______. (Assume same retardation is produced by brakes)

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 \( 5 \) and \( 10 \) \( \text{s} \), respectively, then the value of \( R \) is ______ \( \text{m} \). [Take \( g = 10 \, \text{m} \, \text{s}^{-2} \)]

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:

A body of mass $2 \text{ kg}$ moving with velocity of $\vec{v}_{\text{in}} = 3\hat{i} + 4\hat{j} \text{ m s}^{-1}$ enters into a constant force field of $6 \text{ N}$ directed along positive z-axis. If the body remains in the field for a period of $\frac{5}{3}$ seconds, then velocity of the body when it emerges from force field is:

A projectile is launched at an angle ‘\( \alpha \)’ with the horizontal with a velocity \( 20 \, \text{m s}^{-1} \). After \( 10 \, \text{s} \), its inclination with horizontal is ‘\( \beta \)’. The value of \( \tan \beta \) will be: \( ( \, g = 10 \, \text{m /s}^{2} \, ) \)

A projectile is projected with velocity of \( 25 \, \text{m/s} \) 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 \( g = 10 \, \text{m/s}^{2} \)]

A gun mounted on the ground fires bullets in all directions with same speed. The farthest distance the bullets could reach is \( 64 \, \text{m} \). The speed of the bullets from the gun is _____ \( \text{m/s} \). (take \( g = 10 \, \text{m/s}^2 \))

Two trains '\( A \)' and '\( B \)' of length '\( l \) 'and '\( 4l \)' are travelling into a tunnel of length '\( L \)' in parallel tracks from opposite directions with velocities \( 108 \, \text{km/h} \) and \( 72 \, \text{km/h} \), respectively. If train' \( A \)' take \( 35 \, \text{s} \) less time than train '\( B \)' to cross the tunnel then, length '\( L \)' of tunnel is: (Given \( L = 60 \, l \))

A person can throw a ball up to a maximum range of \( 100 \, \text{m} \). How high above the ground he can throw the same ball?

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

The initial speed of a projectile fired from ground is \( u \). 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 balls with same mass and initial velocity, are projected at different angles in such a way that maximum height reached by first ball is \( 8 \) times higher than that of the second ball. \( T_1 \) and \( T_2 \) are the total flying times of first and second ball, respectively, then the ratio of \( T_1 \) and \( T_2 \) is:

A particle is moving in a circle of radius \( 50 \, \text{cm} \) in such a way that at any instant the normal and tangential components of it's acceleration are equal. If its speed at \( t = 0 \) is \( 4 \, \text{m/s} \), the time taken to complete the first revolution will be \( \dfrac{1}{\alpha} \big[ 1 - e^{-2\pi} \big] \, \text{s} \). where \( \alpha = \) __________.

An object is projected with kinetic energy \( K \) from a point A at an angle \( 60^\circ \) with the horizontal. The ratio of the difference in kinetic energies at points B and C to that at point A (see figure), in the absence of air friction is:

A particle is thrown horizontally from the top of a tower of height \( H \). The angle made by velocity of particle just before hitting the ground is \( 45^\circ \) with the horizontal. What is the horizontal range of particle?

The maximum vertical height to which a man can throw a ball is \( 136 \, \text{m} \). The maximum horizontal distance upto which he can throw the same ball is:

A boy throws a ball into air at \( 45^\circ \) from the horizontal to land it on a roof of a building of height \( H \). If the ball attains maximum height in \( 2 \, \text{s} \) and lands on the building in \( 3 \, \text{s} \) after launch, then value of \( H \) is _______ \( \text{m} \). (\( g = 10 \, \text{m/s}^2 \))

Motion of a particle in \( x-y \) plane is described by a set of following equations \( x=4\sin\left(\frac{\pi}{2}-\omega t\right)\, \text{m} \) and \( y=4\sin(\omega t)\, \text{m} \). The path of the particle will be:

A particle is moving with constant speed in a circular path. When the particle turns by an angle \( 90^\circ \), the ratio of instantaneous velocity to its average velocity is \( \pi : x\sqrt{2} \). The value of \( x \) will be:

The maximum heights of a projectile for two complementary angles of projection are \( 50 \, \text{m} \) and \( 30 \, \text{m} \) respectively. The initial speed of projectile is: \( (g = 10 \, \text{m/s}^2) \)

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

If the initial velocity in horizontal direction of a projectile is unit vector \( \hat{i} \) and the equation of trajectory is \( y = 5x(1 - x) \). The y component vector of the initial velocity is ________ \( \hat{j} \).
(Take \( g = 10 \, \text{m}/\text{s}^2 \))

Two particles A and B are thrown with initial velocities of \( 40 \, \text{m/s} \) and \( 60 \, \text{m/s} \) at angles \( 30^\circ \) and \( 60^\circ \) with the horizontal respectively. The ratio of their ranges respectively is:
\( (g = 10 \, \text{m/s}^2) \)

A force on an object of mass \( 100 \, \text{g} \) is \( (10\hat{i} + 5\hat{j}) \, \text{N} \). The position of that object at \( t = 2 \, \text{s} \) is \( (a\hat{i} + b\hat{j}) \, \text{m} \) after starting from rest. The value of \( \frac{a}{b} \) will be __________________.

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 angular speed of truck wheel is increased from \( 900 \, \text{rpm} \) to \( 2460 \, \text{rpm} \) in \( 26 \) seconds. The number of revolutions by the truck engine during this time is _______. (Assuming the acceleration to be uniform).

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

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

A body rotating with an angular speed of \( 600 \, \text{rpm} \) is uniformly accelerated to \( 1800 \, \text{rpm} \) in \( 10 \, \text{s} \). The number of rotations made in the process is \( \_\_\_\_\_\_ \).

An object 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.

Projectiles A and B are thrown at angles of \( 45^\circ \) and \( 60^\circ \) with vertical respectively from top of a \( 400 \, \text{m} \) high tower. If their range and times of flight are same, the ratio of their speeds of projection \( v_A : v_B \) is:
[Take \( g = 10 \, \text{m s}^{-2} \) ]

The ranges and heights for two projectiles projected with the same initial velocity at angles \( 42^\circ \) and \( 48^\circ \) with the horizontal are \( R_{1}, R_{2} \) and \( H_{1}, H_{2} \) respectively. Choose the correct option:

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

The projectile motion of a particle of mass \( 5 \, \text{g} \) is shown in the figure.
The initial velocity of the particle is \( 5\sqrt{2} \, \text{m s}^{-1} \) and the air resistance is assumed to be negligible. The magnitude of the change in momentum between the points A and B is \( x \times 10^{-2} \, \text{kg m s}^{-1} \). The value of \( x \), to the nearest integer, 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:

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

When one moves from a point \( 16 \, \text{km} \) below the earth's surface to a point \( 16 \, \text{km} \) above the earth's surface. The change in \( g \) is approximately \( \alpha\% \). The value of \( \alpha \) is ______.

A wedge \( Y \) with mass of \( 10 \, \text{kg} \) and all frictionless surfaces and the inclined surface making \( 37^\circ \) with horizontal. A block \( X \) with mass \( 2 \, \text{kg} \) is placed at the highest point of the wedge as shown in figure is at rest. At \( t = 0 \) wedge \( (Y) \) is pulled toward right with constant force \( (f) \) of \( 24 \, \text{N} \). Taking the block \( X \) at rest at \( t = 0 \), the time taken by it to slide down \( 8.8 \, \text{m} \) on the slope, while \( Y \) is on the move, is _____ \( \text{s} \).
(take \( \tan(37^\circ) = \frac{3}{4} \) and \( g = 10 \, \text{m}/\text{s}^2 \) )

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

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:

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:

A river of width \( 200 \, \text{m} \) is flowing from west to east with a speed of \( 18 \, \text{km/h} \). A boat, moving with speed of \( 36 \, \text{km/h} \) 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.

If a particle is projected from origin and it follows the trajectory \( y=x-\dfrac{1}{2}x^2 \), then the time of flight is:
\( (g=\text{acceleration due to gravity}) \)

The maximum height reached by a projectile is \( 64 \, \text{m} \). If the initial velocity is halved, the new maximum height of the projectile is _____ \( \text{m} \).

A $4\text{ kg}$ mass moves under the influence of a force $\vec{F} = \left( 4t^3\hat{i} - 3t\hat{j} \right)\text{N}$ where $t$ is the time in second. If mass starts from origin at $t = 0$, the velocity and position after $t = 2\text{s}$ will be:

A huge circular arc of length \( 4.4 \, \text{ly} \) subtends an angle ‘\( 4s \)’ at the centre of the circle. How long it would take for a body to complete \( 4 \) revolution if its speed is \( 8 \, \text{AU per second} \)?
Given:
\( 1 \, \text{ly} = 9.46 \times 10^{15} \, \text{m} \)
\( 1 \, \text{AU} = 1.5 \times 10^{11} \, \text{m} \)

A projectile is thrown up from top of a tower of height \( 30 \, \text{m} \) at angle \( 30^\circ \) with horizontal, with speed of \( 10\sqrt{3} \, \text{m/s} \). Based on above information match the entries of Column I, with Column – II. \( \\text{(take } g = 10 \, \text{m/s}^2 \\text{)} \)

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.)

A helicopter is flying horizontally with a speed \( v \) at an altitude \( h \) has to drop a food packet for a man on the ground. What is the distance of helicopter from the man when the food packet is dropped?

A helicopter flying horizontally with a speed of \( 360 \, \text{km}/\text{h} \) at an altitude of \( 2 \, \text{km} \), drops an object at an instant. The object hits the ground at a point O, \( 20 \, \text{s} \) after it is dropped. Displacement of ‘O’ from the position of helicopter where the object was released is:
(use acceleration due to gravity \( g = 10 \, \text{m}/\text{s}^2 \) and neglect air resistance)

Angular displacement of a particle moving along a circle of radius \( 2 \, \text{m} \) varies as \( \theta = t^2 \, \text{rad} \). Find the centripetal acceleration of the particle at time \( t = 3 \, \text{sec} \).

Three masses \( m_{1} = 4 \, \text{kg} \), \( m_{2} = 4 \, \text{kg} \) and \( m_{3} = 6 \, \text{kg} \) are suspended from a fixed smooth frictionless pully as shown in the figure below. The value of \( \dfrac{T_{1}}{T_{2}} \) is _______.

\( \text{(take } g = 10 \, \text{m/s}^{2} ) \)

A ball is projected from the ground with a speed \( 15 \, \text{ms}^{-1} \) at an angle \( \theta \) with horizontal so that its range and maximum height are equal, then ‘\( \tan \theta \)’ will be equal to:

A bead \( P \) sliding on a frictionless semi-circular string \( (ACB) \) and it is at point \( S \) at \( t = 0 \) and at this instant the horizontal component of its velocity is \( v \). Another bead \( Q \) of the same mass as \( P \) is ejected from point \( A \) at \( t = 0 \) along the horizontal string \( AB \), with the speed \( v \), friction between the beads and the respective strings may be neglected in both cases. Let \( t_{p} \) and to be the respective times taken by beads \( P \) and \( Q \) to reach the point \( B \), then the relation between \( t_{p} \) and \( t_{Q} \) is:

A particle starts from origin at $t = 0$ with a velocity $5 \hat{i}\text{ m/s}$ and moves in x-y plane under action of a force which produces a constant acceleration of $(3\hat{i} + 2\hat{j})\text{ m/s}^2$. If the x-coordinate of the particle at that instant is $84\text{ m}$, then the speed of the particle at this time is $\sqrt{\alpha}\text{ m/s}$. The value of $\alpha$ is ___________.

A particle moving in a circle of radius \( R \) with uniform speed takes time \( T \) to complete one revolution. If this particle is projected with the same speed at an angle \( \theta \) to the horizontal, the maximum height attained by it is equal to \( 4R \). The angle of projection \( \theta \) is then given by:

A ball of mass \( m \) is thrown vertically upward. Another ball of mass \( 2m \) is thrown at an angle \( \theta \) with the vertical. Both the balls stay in air for the same period of time. The ratio of the heights attained by the two balls respectively is \( {1}/{x} \). The value of \( x \) is _______.

Given below are two statements: One is labelled as Assertion (A) and other is labelled as Reason (R).
Initial velocity \( \vec{u} \) and acceleration \( \vec{a} \) of a particle are given as \( \vec{u}=2\hat{i}+5\hat{j} \, \text{m/s} \; ; \; \vec{a}=-5\hat{i} \, \text{m/s}^2 \)
Assertion (A) : Trajectory of particle is parabolic.
Reason (R) : If motion in \( 2 \)-Dimension is uniform and uniformly accelerated along mutually perpendicular directions, then path followed is parabolic.
In the light of above statements, choose the correct answer from the options given below.

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:

The maximum height of a projectile for two complementary angles of projection are \( 50 \, \text{m} \) and \( 30 \, \text{m} \) respectively. The initial speed of projectile is: \( (g = 10 \, \text{m}/\text{s}^2) \)

A fly wheel is accelerated uniformly from rest and rotates through \( 5 \, \text{rad} \) in the first second. The angle rotated by the fly wheel in the next second, will be:

The angle of projection of a particle is measured from the vertical axis as \( \phi \) and the maximum height reached by the particle is \( h_{m} \). Here \( h_{m} \) as function of \( \phi \) can be presented as:

A fighter jet is flying horizontally at a certain altitude with a speed of \( 200 \, \text{ms}^{-1} \). When it passes directly overhead an anti-aircraft gun, a bullet is fired from the gun, at an angle \( \theta \) with the horizontal, to hit the jet. If the bullet speed is \( 400 \, \text{m/s} \), the value of \( \theta \) will be ____________\( ^\circ \).

A particle is rotating in a circular path and at any instant its motion can be described as \( \theta = \frac{5 t^4}{40} - \frac{t^3}{3} \). The angular acceleration of the particle after \( 10 \) seconds is ______ \( \text{rad}/\text{s}^2 \).

A body of mass \( M \) thrown horizontally with velocity \( v \) from the top of the tower of height \( H \) touches the ground at a distance of \( 100 \, \text{m} \) from the foot of the tower. A body of mass \( 2M \) thrown at a velocity \(\displaystyle  \frac{v}{2} \) from the top of the tower of height \( 4H \) will touch the ground at a distance of __________ \( \text{m} \).

A man carrying a monkey on his shoulder does cycling smoothly on a circular track of radius \( 9 \, \text{m} \) and completes \( 120 \) resolutions in \( 3 \) minutes. The magnitude of centripetal acceleration of monkey is (in \( {m} \, /{s}^{-2} \)):

Angular displacement of a particle moving along a circle of radius \( 2 \, \text{m} \) varies as \( \theta = t^2 \, \text{rad} \). Find the centripetal acceleration of the particle at time \( t = 3 \, \text{sec} \).

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

A body is moving with constant speed, in a circle of radius \( 10 \, \text{m} \). The body completes one revolution in \( 4 \, \text{s} \). At the end of \( 3^{\text{rd}} \) second, the displacement of body (in \( \text{m} \)) from its starting point is:

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 \( \dfrac{\sqrt{2}\, m u^{3}}{X g} \). The value of ‘\( X \)’ is ______________.

A ball having kinetic energy \( KE \), is projected at an angle of \( 60^\circ \) from the horizontal. What will be the kinetic energy of ball at the highest point of its flight?

A bomb is dropped by a fighter plane flying horizontally. To an observer sitting in the plane, the trajectory of the bomb is a: