## Numerical

For a train engine moving with speed of $$20 \mathrm{~ms}^{-1}$$, the driver must apply brakes at a distance of 500 $$\mathrm{m}$$ before the station ...

Two bodies are projected from ground with same speeds $40 \mathrm{~ms}^{-1}$ at two different angles with respect to horizontal. The bodies were found...

The speed of a swimmer is $$4 \mathrm{~km} \mathrm{~h}^{-1}$$ in still water. If the swimmer makes his strokes normal to the flow of river of width $$...

A horse rider covers half the distance with $$5 \mathrm{~m} / \mathrm{s}$$ speed. The remaining part of the distance was travelled with speed $$10 \ma...

A tennis ball is dropped on to the floor from a height of 9.8 m. It rebounds to a height 5.0 m. Ball comes in contact with the floor for 0.2s. The ave...

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 maximu...

A ball is thrown vertically upwards with a velocity of $$19.6 \mathrm{~ms}^{-1}$$ from the top of a tower. The ball strikes the ground after $$6 \math...

Two inclined planes are placed as shown in figure. A block is projected from the Point A of inclined plane AB along its surface with a velocity just s...

A ball of mass m is thrown vertically upward. Another ball of mass $$2 \mathrm{~m}$$ is thrown at an angle $$\theta$$ with the vertical. Both the ball...

If the initial velocity in horizontal direction of a projectile is unit vector $$\hat{i}$$ and the equation of trajectory is $$y=5 x(1-x)$$. The $$y$$...

A particle is moving in a straight line such that its velocity is increasing at 5 ms$$-$$1 per meter. The acceleration of the particle is ____________...

A car is moving with speed of $$150 \mathrm{~km} / \mathrm{h}$$ and after applying the break it will move $$27 \mathrm{~m}$$ before it stops. If the s...

A car covers AB distance with first one-third at velocity v1 ms$$-$$1, second one-third at v2 ms$$-$$1 and last one-third at v3 ms$$-$$1. If v3 = 3v1,...

A ball is projected vertically upward with an initial velocity of 50 ms$$-$$1 at t = 0s. At t = 2s, another ball is projected vertically upward with s...

A fighter jet is flying horizontally at a certain altitude with a speed of 200 ms$$-$$1. When it passes directly overhead an anti-aircraft gun, a bull...

A ball of mass 0.5 kg is dropped from the height of 10 m. The height, at which the magnitude of velocity becomes equal to the magnitude of acceleratio...

A body is projected from the ground at an angle of 45$$^\circ$$ with the horizontal. Its velocity after 2s is 20 ms$$-$$1. The maximum height reached ...

From the top of a tower, a ball is thrown vertically upward which reaches the ground in 6 s. A second ball thrown vertically downward from the same po...

A particle is moving with constant acceleration 'a'. Following graph shows v2 versus x(displacement) plot. The acceleration of the particle is _______...

If the velocity of a body related to displacement x is given by $$\upsilon = \sqrt {5000 + 24x} $$ m/s, then the acceleration of the body is ...........

Two spherical balls having equal masses with radius of 5 cm each are thrown upwards along the same vertical direction at an interval of 3s with the sa...

A swimmer wants to cross a river from point A to point B. Line AB makes an angle of 30$$^\circ$$ with the flow of river. Magnitude of velocity of the ...

Three particles P, Q and R are moving along the vectors $$\overrightarrow A = \widehat i + \widehat j$$, $$\overrightarrow B = \widehat j + \widehat...

A person is swimming with a speed of 10 m/s at an angle of 120$$^\circ$$ with the flow and reaches to a point directly opposite on the other side of t...

A swimmer can swim with velocity of 12 km/h in still water. Water flowing in a river has velocity 6 km/h. The direction with respect to the direction ...

The speed verses time graph for a particle is shown in the figure. The distance travelled (in m) by
the particle during the time interval t = 0 to t =...

The distance x covered by a particle in one
dimensional motion varies with time t as
x2 = at2 + 2bt + c. If the acceleration of the
particle depends o...

A ball is dropped from the top of a 100 m high
tower on a planet. In the last $${1 \over 2}s$$ before hitting
the ground, it covers a distance of 19 m...

A particle is moving along the x-axis with its
coordinate with the time 't' given be x(t) = 10 + 8t – 3t2. Another particle is moving
the y-axis with ...

## MCQ (Single Correct Answer)

An object moves with speed $$v_1,v_2$$ and $$v_3$$ along a line segment AB, BC and CD respectively as shown in figure. Where AB = BC and AD = 3AB, the...

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}^...

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{\sqr...

Spherical insulating ball and a spherical metallic ball of same size and mass are dropped from the same height. Choose the correct statement out of th...

A vehicle travels $4 \mathrm{~km}$ with speed of $3 \mathrm{~km} / \mathrm{h}$ and another $4 \mathrm{~km}$ with speed of $5 \mathrm{~km} / \mathrm{h}...

Match Column-I with Column-II :
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The distance travelled by a particle is related to time t as $$x=4\mathrm{t}^2$$. The velocity of the particle at t=5s is :-

Two objects are projected with same velocity 'u' however at different angles $$\alpha$$ and $$\beta$$ with the horizontal. If $$\alpha+\beta=90^\circ$...

A car travels a distance of '$$x$$' with speed $$v_1$$ and then same distance '$$x$$' with speed $$v_2$$ in the same direction. The average speed of t...

The velocity time graph of a body moving in a straight line is shown in the figure.
The ratio of displacement to distance travelled by the body in ti...

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 :...

A juggler throws balls vertically upwards with same initial velocity in air. When the first ball reaches its highest position, he throws the next ball...

A ball is released from a height h. If $$t_{1}$$ and $$t_{2}$$ be the time required to complete first half and second half of the distance respectivel...

A ball is thrown up vertically with a certain velocity so that, it reaches a maximum height h. Find the ratio of the times in which it is at height $$...

If $$\mathrm{t}=\sqrt{x}+4$$, then $$\left(\frac{\mathrm{d} x}{\mathrm{~d} t}\right)_{\mathrm{t}=4}$$ 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...

A NCC parade is going at a uniform speed of $$9 \mathrm{~km} / \mathrm{h}$$ under a mango tree on which a monkey is sitting at a height of $$19.6 \mat...

The velocity of the bullet becomes one third after it penetrates 4 cm in a wooden block. Assuming that bullet is facing a constant resistance during i...

A bullet is shot vertically downwards with an initial velocity of $$100 \mathrm{~m} / \mathrm{s}$$ from a certain height. Within 10 s, the bullet reac...

Two projectiles are thrown with same initial velocity making an angle of $$45^{\circ}$$ and $$30^{\circ}$$ with the horizontal respectively. The rati...

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

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, th...

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...

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 an...

A small toy starts moving from the position of rest under a constant acceleration. If it travels a distance of 10m in t s, the distance travelled by t...

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

Two balls A and B are placed at the top of 180 m tall tower. Ball A is released from the top at t = 0 s. Ball B is thrown vertically down with an init...

Velocity (v) and acceleration (a) in two systems of units 1 and 2 are related as $${v_2} = {n \over {{m^2}}}{v_1}$$ and $${a_2} = {{{a_1}} \over {mn}}...

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 '$$\...

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 spee...

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...

Two buses P and Q start from a point at the same time and move in a straight line and their positions are represented by $${X_P}(t) = \alpha t + \beta...

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 ...

The ranges and heights for two projectiles projected with the same initial velocity at angles 42$$^\circ$$ and 48$$^\circ$$ with the horizontal are R1...

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 hel...

Water drops are falling from a nozzle of a shower onto the floor, from a height of 9.8 m. The drops fall at a regular interval of time. When the first...

A player kicks a football with an initial speed of 25 ms$$-$$1 at an angle of 45$$^\circ$$ from the ground. What are the maximum height and the time t...

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

A ball is thrown up with a certain velocity so that it reaches a height 'h'. Find the ratio of the two different times of the ball reaching $${h \over...

The relation between time t and distance x for a moving body is given as t = mx2 + nx, where m and n are constants. The retardation of the motion is :...

A balloon was moving upwards with a uniform velocity of 10 m/s. An object of finite mass is dropped from the balloon when it was at a height of 75 m f...

The instantaneous velocity of a particle moving in a straight line is given as $$V = \alpha t + \beta {t^2}$$, where $$\alpha$$ and $$\beta$$ are cons...

Water droplets are coming from an open tap at a particular rate. The spacing between a droplet observed at 4th second after its fall to the next dropl...

A boy reaches the airport and finds that the escalator is not working. He walks up the stationary escalator in time t1. If he remains stationary on a ...

A butterfly is flying with a velocity $$4\sqrt 2 $$ m/s in North-East direction. Wind is slowly blowing at 1 m/s from North to South. The resultant di...

The velocity $$-$$ displacement graph of a particle is shown in the figure.The acceleration $$-$$ displacement graph of the same particle is represent...

The position, velocity and acceleration of a particle moving with a constant acceleration can be represented by :

The velocity of a particle is v = v0 + gt + Ft2. Its position is x = 0 at t = 0; then its displacement after time (t = 1) is :

A rubber ball is released from a height of 5 m above the floor. It bounces back repeatedly, always rising to $${{81} \over {100}}$$ of the height thro...

A car accelerates from rest at a constant rate $$\alpha$$ for some time after which it decelerates at a constant rate $$\beta$$ to come to rest. If th...

A mosquito is moving with a velocity $$\overrightarrow v = 0.5{t^2}\widehat i + 3t\widehat j + 9\widehat k$$ m/s and accelerating in uniform conditio...

The velocity-displacement graph describing the motion of bicycle is shown in the figure.The acceleration-displacement graph of the bicycle's motion is...

A scooter accelerates from rest for time t1 at constant rate a1 and then retards at constant rate a2 for time t2 and comes to rest. The correct value ...

The trajectory of a projectile in a vertical plane is y = $$\alpha$$x $$-$$ $$\beta$$x2, where $$\alpha$$ and $$\beta$$ are constants and x & y ar...

A stone is dropped from the top of a building. When it crosses a point 5 m below the top, another stone starts to fall from a point 25 m below the top...

An engine of a train, moving with uniform acceleration, passes the signal-post with velocity u and the last compartment with velocity v. The velocity ...

If the velocity-time graph has the shape AMB, what would be the shape of the corresponding acceleration-time graph?...

When a car is at rest, its driver sees rain drops falling on it vertically. When driving the car with
speed v, he sees that rain drops are coming at a...

The velocity (v) and time (t) graph of a body in
a straight line motion is shown in the figure.
The point S is at 4.333 seconds. The total
distance co...

A helicopter rises from rest on the ground
vertically upwards with a constant acceleration
g. A food packet is dropped from the helicopter
when it is ...

A balloon is moving up in air vertically above a
point A on the ground. When it is at a height h1,
a girl standing at a distanced (point B) from A
(se...

A Tennis ball is released from a height h and after freely falling on a wooden floor
it rebounds and reaches height $${h \over 2}$$. The
velocity vers...

Starting from the origin at time t = 0, with initial velocity 5$$\widehat j$$ ms-1 , a particle moves in the x-y plane
with a constant acceleration of...

Train A and train B are running on parallel
tracks in the opposite directions with speeds of
36 km/hour and 72 km/hour, respectively. A
person is walk...

A particle starts from the origin at t = 0 with an
initial velocity of 3.0 $$\widehat i$$ m/s and moves in the
x-y plane with a constant acceleration
...

A particle moves such that its position
vector $$\overrightarrow r \left( t \right) = \cos \omega t\widehat i + \sin \omega t\widehat j$$ where $$\ome...

A particle is moving with speed v = b$$\sqrt x $$ along positive x-axis. Calculate the speed of the particle at
time t = $$\tau $$(assume that the par...

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, h1 and h2. ...

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 t1 and t2 a...

The trajectory of a projectile near the surface of the earth is given as y = 2x – 9x2
. If it were launched at an
angle $$\theta $$0 with speed v0 the...

A plane is inclined at an angle $$\alpha $$ = 30° with respect to the horizontal. A particle is projected with a speed u =
2 ms–1
, from the base of t...

A ball is thrown upward with an initial velocity
V0 from the surface of the earth. The motion
of the ball is affected by a drag force equal to
m$$\gam...

The position of a particle as a function of time
t, is given byx(t) = at + bt2 – ct3
where a, b and c are constants. When the
particle attains zero ac...

The position vector of a particle changes with
time according to the relation
$$\overrightarrow r (t) = 15{t^2}\widehat i + (4 - 20{t^2})\widehat j$$ ...

The stream of a river is flowing with a speed
of 2km/h. A swimmer can swim at a speed of
4km/h. What should be the direction of the
swimmer with respe...

A particle starts from origin O from rest and
moves with a uniform acceleration along the
positive x-axis. Identify all figures that
correctly represe...

Ship A is sailing towards north-east with
velocity $$\mathop v\limits^ \to = 30\mathop i\limits^ \wedge + 50\mathop j\limits^ \wedge $$ km/hr whe...

A person standing on an open ground hears the sound of a jet aeroplane, coming from north at an angle 60o with ground level. But he finds the aeroplan...

A passenger train of length 60 m travels at a speed of 80 km/hr. Another freight train of length 120 m travels at a speed of 30 km/hr. The ratio of ti...

A particle moves from the point $$\left( {2.0\widehat i + 4.0\widehat j} \right)$$ m, at t = 0, with an initial velocity $$\left( {5.0\widehat i + 4.0...

A particle starts from the origin at time t = 0 and moves along the positive x-axis. The graph of velocity with respect to time is shown in figure. Wh...

Two guns A and B can fire bullets at speeds 1 km/s and 2 km/s respectively. From a point on a horizontal ground, they are fired in all possible direct...

In a car race on straight road, car A takes a time t less than car B at the finish and passes finishing point with a speed '$$\upsilon $$' more than t...

The position co-ordinates of a particle moving in a 3-D coordinate system is given by
x = a cos$$\omega $$t
y = a sin$$\omega $$t and
z = a$$\omega ...

A particle is moving with a velocity
$$\overrightarrow v \, = K(y\widehat i + x\widehat j),$$ where K is a constant.
The general equation for its pat...

All the graphs below are intended to represent the same motion. One of them does it incorrectly. Pick it up.

A man in a car at location Q on a straight highway is moving with speed $$\upsilon $$. He decides to reach a point P in a field at a distance d from t...

An automobile, travelling at $$40\,$$ km/h, can be stopped at a distance of $$40\,$$ m by applying brakes. If the same automobile is travelling at $$8...

The velocity-time graphs of a car and a scooter are shown in the figure. (i) The difference between the distance travelled by the car and the scooter ...

A car is standing 200 m behind a bus, which is also at rest. The two start moving at the same instant but with different forward accelerations. The bu...

Which graph corresponds to an object moving with a constant negative
acceleration and a positive velocity ?

A body is thrown vertically upwards. Which one of the following graphs correctly represent the velocity vs
time?

Two stones are thrown up simultaneously from the edge of a cliff $$240$$ $$m$$ high with initial speed of $$10$$ $$m/s$$ and $$40$$ $$m/s$$ respective...

From a tower of height H, a particle is thrown vertically upwards with a speed u. The time taken by the
particle, to hit the ground, is n times that t...

A projectile is given an initial velocity of $$\left( {\widehat i + 2\widehat j} \right)$$ m/s, where $${\widehat i}$$ is along the ground and $${\wid...

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

An object, moving with a speed of 6.25 m/s, is decelerated at a rate given by :
$${{dv} \over {dt}} = - 2.5\sqrt v $$ where v is the instantaneous sp...

A water fountain on the ground sprinkles water all around it. If the speed of water coming out of the
fountain is v, the total area around the fountai...

A particle is moving with velocity $$\overrightarrow v = k\left( {y\widehat i + x\widehat j} \right)$$, where K is a constant. The general equation f...

A small particle of mass $$m$$ is projected at an angle $$\theta $$ with the $$x$$-axis with an initial velocity $${v_0}$$ in the $$x$$-$$y$$ plane as...

A particle has an initial velocity $$3\widehat i + 4\widehat j$$ and an acceleration of $$0.4\widehat i + 0.3\widehat j$$. Its speed after 10 s is:

Consider a rubber ball freely falling from a height $$h=4.9$$ $$m$$ onto a horizontal elastic plate. Assume that the duration of collision is negligi...

A body is at rest at $$x=0.$$ At $$t=0,$$ it starts moving in the positive $$x$$-direction with a constant acceleration. At the same instant another b...

The velocity of a particle is v = v0 + gt + ft2. If its position is x = 0 at t = 0, then its displacement after unit time (t = 1) is...

A particle located at x = 0 at time t = 0, starts moving along the positive x-direction with a velocity
'v' that varies as $$v = \alpha \sqrt x $$. Th...

A car starting from rest accelerates at the rate f through a distance S, then continues
at constant speed for time t and then decelerates at the rate ...

The relation between time t and distance x is t = ax2 + bx where a and b are constants.
The acceleration is

A particle is moving eastwards with a velocity of 5 m/s. In 10 seconds the velocity
changes to 5 m/s northwards. The average acceleration in this time...

A ball is released from the top of a tower of height h meters. It takes T seconds to reach the
ground. What is the position of the ball in $${T \over ...

A projectile can have the same range 'R' for two angles of projection. If T1 and T2 be the time
of flights in the two cases, then the product of the t...

An automobile travelling with speed of 60 km/h, can brake to stop within a distance of 20 m.
If the car is going twice as fast, i.e 120 km/h, the stop...

A ball is thrown from a point with a speed ν0 at an angle of projection θ. From the same point
and at the same instant person starts running with a co...

A car, moving with a speed of 50 km/hr, can be stopped by brakes after at least 6 m. If the
same car is moving at a speed of 100 km/hr, the minimum st...

A boy playing on the roof of a 10 m high building throws a ball with a speed of 10 m/s at an
angle of $$30^\circ $$ with the horizontal. How far from ...

The co-ordinates of a moving particle at any time 't' are given by x = $$\alpha $$t3 and y = βt3. The speed to the particle at time 't' is given by...

From a building two balls A and B are thrown such that A is thrown upwards and B downwards ( both vertically with the same speed ). If vA and vB are ...

Speeds of two identical cars are $$u $$ and $$4$$$$u $$ at the specific instant. The ratio of the respective distances in which the two cars are stopp...