LFM Pure and Mechanics

iran-konkur 2013 Q159 View

159- A ball is thrown vertically upward from a height of 20 m above the ground with initial speed $V_0$. It reaches a height of 65 m above the ground. The speed of the ball becomes zero at ground level. If $g = 10\,\dfrac{\text{m}}{\text{s}^2}$, how many meters per second is $V_0$?
(Air resistance is negligible)
(1) $35$ (2) $30$ (3) $13\sqrt{10}$ (4) $10\sqrt{13}$

iran-konkur 2013 Q160 View

160- The initial velocity vector of a projectile in SI is $\vec{V}_0 = 15\vec{i} + 20\vec{j}$. What is the displacement vector of this projectile in the first 3 seconds in SI?
($g = 10\,\dfrac{\text{m}}{\text{s}^2}$ and air resistance is negligible.)
(1) $45\vec{i} + 15\vec{j}$ (2) $15\vec{i} - 10\vec{j}$ (3) $45\vec{i} - 10\vec{j}$ (4) $10\vec{i} + 45\vec{j}$

iran-konkur 2019 Q158 View

158. A bullet is fired from height $h$ with speed $V$. The bullet passes through the ground at height $9\,\text{m}$ and reaches the ground with speed $\dfrac{3}{2}V$. $h$ is how many meters? (Ignore air resistance and $g = 10\,\dfrac{\text{m}}{\text{s}^2}$)
(1) $16.2$ (2) $18$ (3) $33.4$ (4) $36$

iran-konkur 2019 Q162 View

162. A satellite of mass $500\,\text{kg}$ orbits the Earth at an altitude of $1600\,\text{km}$ above the Earth's surface. How many newtons of gravitational force acts on the satellite? $$\left(g = 10\,\frac{\text{m}}{\text{s}^2} \text{ and } R_e = 6400\,\text{km}\right)$$

[(1)] $5000$
[(2)] $3700$
[(3)] $800$
[(4)] $640$

jee-advanced 2021 Q5 2 marks Projectile with Mid-Flight Event (Breakup or Bounce) View

A projectile is thrown from a point O on the ground at an angle $45^\circ$ from the vertical and with a speed $5\sqrt{2}$ m/s. The projectile at the highest point of its trajectory splits into two equal parts. One part falls vertically down to the ground, 0.5 s after the splitting. The other part, $t$ seconds after the splitting, falls to the ground at a distance $x$ meters from the point O. The acceleration due to gravity $g = 10$ m/s$^2$.
The value of $t$ is ____.

jee-main 2003 Q7 Relative velocity and observed length/time View

Two particles start simultaneously from the same point and move along two straight lines, one with uniform velocity $\overrightarrow{\mathrm{u}}$ and the other from rest with uniform acceleration $\overrightarrow{\mathrm{f}}$. Let $\alpha$ be the angle between their directions of motion. The relative velocity of the second particle w.r.t. the first is least after a time.
(1) $\frac{u\cos\alpha}{f}$
(2) $\frac{u\sin\alpha}{f}$
(3) $\frac{f\cos\alpha}{u}$
(4) $u\sin\alpha$

jee-main 2004 Q5 Finding Angle of Projection from Given Conditions View

A ball is thrown from a point with a speed $v _ { 0 }$ at an angle of projection $\theta$. From the same point and at the same instant person starts running with a constant speed $v _ { 0 } / 2$ to catch the ball. Will the person be able to catch the ball? If yes, what should be the angle of projection?
(1) yes, $60 ^ { \circ }$
(2) yes, $30 ^ { \circ }$
(3) no
(4) yes, $45 ^ { \circ }$

jee-main 2004 Q6 Range and Complementary Angle Relationships View

A projectile can have the same range $R$ for two angles of projection. If $T _ { 1 }$ and $T _ { 2 }$ be the time of flights in the two cases, then the product of the two time of flights is directly proportional to
(1) $1 / R ^ { 2 }$
(2) $1 / R$
(3) R
(4) $R ^ { 2 }$

jee-main 2004 Q7 Force from velocity change (Newton's second law with impulse/momentum) View

If $t _ { 1 }$ and $t _ { 2 }$ are the times of flight of two particles having the same initial velocity $u$ and range R on the horizontal, then $t _ { 1 } ^ { 2 } + t _ { 2 } ^ { 2 }$ is equal to
(1) $\frac { u ^ { 2 } } { g }$
(2) $\frac { 4 u ^ { 2 } } { g ^ { 2 } }$
(3) $\frac { u ^ { 2 } } { 2 g }$
(4) 1

jee-main 2005 Q1 Vector Word Problem / Physical Application View

A particle is moving eastwards with a velocity of $5 \mathrm{~m}/\mathrm{s}$ in 10 seconds the velocity changes to $5 \mathrm{~m}/\mathrm{s}$ northwards. The average acceleration in this time is
(1) $\frac{1}{\sqrt{2}} \mathrm{~m}/\mathrm{s}^2$ towards north-east
(2) $\frac{1}{2} \mathrm{~m}/\mathrm{s}^2$ towards north.
(3) zero
(4) $\frac{1}{\sqrt{2}} \mathrm{~m}/\mathrm{s}^2$ towards north-west

jee-main 2007 Q99 View

A tower stands at the centre of a circular park. $A$ and $B$ are two points on the boundary of the park such that $A B ( = a )$ subtends an angle of $60 ^ { \circ }$ at the foot of the tower, and the angle of elevation of the top of the tower from $A$ or $B$ is $30 ^ { \circ }$. The height of the tower is
(1) $\frac { 2 a } { \sqrt { 3 } }$
(2) $2 a \sqrt { 3 }$
(3) $\frac { a } { \sqrt { 3 } }$
(4) $a \sqrt { 3 }$

jee-main 2009 Q1 Force-to-acceleration and resulting kinematics View

A particle has an initial velocity $3 \hat { i } + 4 \hat { j }$ and an acceleration of $0.4 \hat { i } + 0.3 \hat { j }$. Its speed after 10 s is
(1) 10 units
(2) $7 \sqrt { 2 }$ units
(3) 7 units
(4) 8.5 units

jee-main 2012 Q2 Maximum Range or Maximum Height from Given Constraints View

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
(1) $20\sqrt{2}$ m
(2) 10 m
(3) $10\sqrt{2}$ m
(4) 20 m

jee-main 2019 Q1 Force-to-acceleration and resulting kinematics View

A particle moves from the point $( 2.0 \hat { i } + 4.0 \hat { j } ) \mathrm { m }$, at $\mathrm { t } = 0$, with an initial velocity $( 5.0 \hat { i } + 4.0 \hat { j } ) \mathrm { ms } ^ { - 1 }$. It is acted upon by a constant force which produces a constant acceleration $( 4.0 \hat { i } + 4.0 \hat { j } ) \mathrm { ms } ^ { - 2 }$. What is the distance of the particle from the origin at time 2 s ?
(1) 15 m
(2) $20 \sqrt { 2 } \mathrm {~m}$
(3) 5 m
(4) $10 \sqrt { 2 } \mathrm {~m}$

jee-main 2020 Q3 View

Starting from the origin at time $\mathrm { t } = 0$, with initial velocity $5 \widehat { \mathrm { j } } \mathrm { ms } ^ { - 1 }$, a particle moves in the $x - y$ plane with a constant acceleration of $( 10 \widehat { \mathrm { i } } + 4 \widehat { \mathrm { j } } ) \mathrm { ms } ^ { - 2 }$. At time t , its coordinates are $\left( 20 \mathrm {~m} , \mathrm { y } _ { 0 } \mathrm {~m} \right)$. The values of t and $\mathrm { y } _ { 0 }$ are, respectively:
(1) 2 s and 18 m
(2) 4 s and 52 m
(3) 2 s and 24 m
(4) 5 s and 25 m

jee-main 2020 Q4 View

The acceleration due to gravity on the earth's surface at the poles is $g$ and angular velocity of the earth about the axis passing through the pole is $\omega$. An object is weighed at the equator and at a height $h$ above the poles by using a spring balance. If the weights are found to be same, then $h$ is: ($h \ll R$, where $R$ is the radius of the earth)
(1) $\frac{R^2\omega^2}{2g}$
(2) $\frac{R^2\omega^2}{g}$
(3) $\frac{R^2\omega^2}{4g}$
(4) $\frac{R^2\omega^2}{8g}$

jee-main 2020 Q5 View

A box weighs 196 N on a spring balance at the north pole. Its weight recorded on the same balance if it is shifted to the equator is close to (Take $\mathrm { g } = 10 \mathrm {~ms} ^ { - 2 }$ at the north pole and the radius of the earth $= 6400 \mathrm {~km}$):
(1) 195.66 N
(2) 194.32 N
(3) 194.66 N
(4) 195.32 N

jee-main 2021 Q2 View

Water droplets are coming from an open tap at a particular rate. The spacing between a droplet observed at $4^{\text{th}}$ second after its fall to the next droplet is 34.3 m. At what rate the droplets are coming from the tap? (Take $g = 9.8 \mathrm{~m~s}^{-2}$)
(1) 3 drops$/2$ seconds
(2) 2 drops$/$ second
(3) 1 drop/second
(4) 1 drop$/7$ seconds

jee-main 2021 Q21 View

A body of mass 2 kg moves under a force of $( 2 \hat { \mathrm { i } } + 3 \hat { \mathrm { j } } + 5 \widehat { \mathrm { k } } ) \mathrm { N }$. It starts from rest and was at the origin initially. After 4 s, its new coordinates are $( 8 , b , 20 )$. The value of $b$ is $\_\_\_\_$. (Round off to the Nearest Integer)

jee-main 2021 Q21 View

A person is swimming with a speed of $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $120 ^ { \circ }$ with the flow and reaches to a point directly opposite on the other side of the river. The speed of the flow is $x \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The value of $x$ to the nearest integer is $\_\_\_\_$.

jee-main 2021 Q21 View

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 the river. The magnitude of the velocity of the swimmer is the same as that of the river. The angle $\theta$ with the line $AB$ should be $\_\_\_\_$ ${ } ^ { \circ }$, so that the swimmer reaches point $B$.

jee-main 2022 Q21 Vertical projection (up or down) from a height View

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 position with the same speed reaches the ground in 1.5 s . A third ball released, from the rest from the same location, will reach the ground in $\_\_\_\_$ s.

jee-main 2022 Q66 Heights and distances / angle of elevation problem View

A horizontal park is in the shape of a triangle $O A B$ with $A B = 16$. A vertical lamp post $O P$ is erected at the point $O$ such that $\angle P A O = \angle P B O = 15 ^ { \circ }$ and $\angle P C O = 45 ^ { \circ }$, where $C$ is the midpoint of $A B$. Then $( O P ) ^ { 2 }$ is equal to
(1) $\frac { 32 } { \sqrt { 3 } } ( \sqrt { 3 } - 1 )$
(2) $\frac { 32 } { \sqrt { 3 } } ( 2 - \sqrt { 3 } )$
(3) $\frac { 16 } { \sqrt { 3 } } ( \sqrt { 3 } - 1 )$
(4) $\frac { 16 } { \sqrt { 3 } } ( 2 - \sqrt { 3 } )$

jee-main 2022 Q69 View

The angle of elevation of the top $P$ of a vertical tower $P Q$ of height 10 from a point $A$ on the horizontal ground is $45 ^ { \circ }$. Let $R$ be a point on $A Q$ and from a point $B$, vertically above $R$, the angle of elevation of $P$ is $60 ^ { \circ }$. If $\angle B A Q = 30 ^ { \circ } , A B = d$ and the area of the trapezium $P Q R B$ is $\alpha$, then the ordered pair ( $d , \alpha$ ) is
(1) $( 10 ( \sqrt { 3 } - 1 ) , 25 )$
(2) $\left( 10 ( \sqrt { 3 } - 1 ) , \frac { 25 } { 2 } \right)$
(3) $( 10 ( \sqrt { 3 } + 1 ) , 25 )$
(4) $\left( 10 ( \sqrt { 3 } + 1 ) , \frac { 25 } { 2 } \right)$

jee-main 2023 Q3 Velocity or Momentum at a Given Time View

A projectile is projected at $30 ^ { \circ }$ from horizontal with initial velocity $40 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The velocity of the projectile at $\mathrm { t } = 2 \mathrm {~s}$ from the start will be:
(1) $40 \sqrt { 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(2) Zero
(3) $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(4) $20 \sqrt { 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$

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LFM Pure and Mechanics

Indices and Surds 96

Exponential Functions 197

Laws of Logarithms 242

Exponential Equations & Modelling 28

Arithmetic Sequences and Series 363

Geometric Sequences and Series 159

Differentiation from First Principles 45

Tangents, normals and gradients 140

Stationary points and optimisation 505

Applied differentiation 76

Indefinite & Definite Integrals 502

Areas Between Curves 71

Areas by integration 221

Volumes of Revolution 68

Numerical integration 49

Vectors Introduction & 2D 211

Vectors 3D & Lines 524

Travel graphs 14

SUVAT in 2D & Gravity 31

Constant acceleration (SUVAT) 78

Variable acceleration (1D) 55

Variable acceleration (vectors) 32

Forces, equilibrium and resultants 24

Newton's laws and connected particles 21

Pulley systems 6

Motion on a slope 6

Friction 8

Momentum and Collisions 34

Moments 28

Parametric curves and Cartesian conversion 9

Parametric differentiation 13

Parametric integration 6

Projectiles 49