LFM Pure and Mechanics

jee-main 2019 Q3 Velocity or Momentum at a Given Time View

A body is projected at $t = 0$ with a velocity $10 \mathrm {~ms} ^ { - 1 }$ at an angle of $60 ^ { \circ }$ with the horizontal. The radius of curvature of its trajectory at $t = 1 \mathrm {~s}$ is $R$. Neglecting air resistance and taking acceleration due to gravity $\mathrm { g } = 10 \mathrm {~ms} ^ { - 2 }$, the value of $R$ is:
(1) 10.3 m
(2) 2.8 m
(3) 2.5 m
(4) 5.1 m

jee-main 2019 Q3 Range and Complementary Angle Relationships View

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 $t_1$ and $t_2$ are the values of the time taken by it to hit the target in two possible ways, the product $t_1 t_2$ is:
(1) $R/2g$
(2) $R/g$
(3) $2R/g$
(4) $R/4g$

jee-main 2019 Q3 Clearing an Obstacle or Passing Through a Point View

A plane is inclined at an angle $\alpha = 30 ^ { \circ }$ with respect to the horizontal. A particle is projected with a speed $\mathrm { u } = 2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, from the base of the plane, making an angle $\theta = 15 ^ { \circ }$ with respect to the plane as shown in the figure. The distance from the base, at which the particle hits the plane is close to: (Take $\mathrm { g } = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ )
(1) 20 cm
(2) 18 cm
(3) 14 cm
(4) 26 cm

jee-main 2019 Q75 Projectile Involving Rotational or Combined Physics View

If the angle of elevation of a cloud from a point $P$ which is $25 m$ above a lake be $30 ^ { \circ }$ and the angle of depression of reflection of the cloud in the lake from $P$ be $60 ^ { \circ }$, then the height of the cloud (in meters) from the surface of the lake is :
(1) 50
(2) 60
(3) 45
(4) 42

jee-main 2021 Q3 Range and Complementary Angle Relationships View

The ranges and heights for two projectiles projected with the same initial velocity at angles $42 ^ { \circ }$ and $48 ^ { \circ }$ with the horizontal are $R _ { 1 } , \quad R _ { 2 }$ and $H _ { 1 } , \quad H _ { 2 }$ respectively. Choose the correct option:
(1) $R _ { 1 } = R _ { 2 }$ and $H _ { 1 } = H _ { 2 }$
(2) $R _ { 1 } = R _ { 2 }$ and $H _ { 1 } < H _ { 2 }$
(3) $R _ { 1 } > R _ { 2 }$ and $H _ { 1 } = H _ { 2 }$
(4) $R _ { 1 } < R _ { 2 }$ and $H _ { 1 } < H _ { 2 }$

jee-main 2021 Q4 Trajectory Equation Analysis View

The trajectory of a projectile in a vertical plane is $y = \alpha x - \beta x ^ { 2 }$, where $\alpha$ and $\beta$ are constants and $x \& y$ are respectively the horizontal and vertical distances of the projectile from the point of projection. The angle of projection $\theta$ and the maximum height attained $H$ are respectively given by
(1) $\tan ^ { - 1 } \alpha , \frac { 4 \alpha ^ { 2 } } { \beta }$
(2) $\tan ^ { - 1 } \left( \frac { \beta } { \alpha } \right) , \frac { \alpha ^ { 2 } } { \beta }$
(3) $\tan ^ { - 1 } \beta , \frac { \alpha ^ { 2 } } { 2 \beta }$
(4) $\tan ^ { - 1 } \alpha , \frac { \alpha ^ { 2 } } { 4 \beta }$

jee-main 2021 Q5 Projectile from a Non-Inertial or Moving Frame View

A bomb is dropped by a fighter plane flying horizontally. To an observer sitting in the plane, the trajectory of the bomb is a:
(1) straight line vertically down the plane
(2) parabola in a direction opposite to the motion of plane
(3) parabola in the direction of motion of plane
(4) hyperbola

jee-main 2021 Q22 Projectile Involving Rotational or Combined Physics View

A swimmer can swim with velocity of $12 \mathrm {~km} / \mathrm { h }$ in still water. Water flowing in a river has velocity $6 \mathrm {~km} / \mathrm { h }$. The direction with respect to the direction of flow of river water he should swim in order to reach the point on the other bank just opposite to his starting point is $\_\_\_\_$. (Round off to the Nearest Integer) (find the angle in degree)

jee-main 2021 Q70 Heights and distances / angle of elevation problem View

A pole stands vertically inside a triangular park $ABC$. Let the angle of elevation of the top of the pole from each corner of the park be $\frac { \pi } { 3 }$. If the radius of the circumcircle of $\triangle ABC$ is 2 , then the height of the pole is equal to:
(1) $\frac { 2 \sqrt { 3 } } { 3 }$
(2) $2 \sqrt { 3 }$
(3) $\sqrt { 3 }$
(4) $\frac { 1 } { \sqrt { 3 } }$

jee-main 2022 Q1 Ratio of Projectile Quantities for Different Launch Parameters View

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
(1) $1 : \sqrt { 2 }$
(2) $\sqrt { 2 } : 1$
(3) $2 : \sqrt { 3 }$
(4) $\sqrt { 3 } : 2$

jee-main 2022 Q2 Finding Angle of Projection from Given Conditions View

A projectile is projected with velocity of $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ 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 $\mathrm { g } = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ ]
(1) $\frac { 1 } { 2 } \sin ^ { - 1 } \left( \frac { 5 t ^ { 2 } } { 4 R } \right)$
(2) $\frac { 1 } { 2 } \sin ^ { - 1 } \left( \frac { 4 R } { 5 t ^ { 2 } } \right)$
(3) $\tan ^ { - 1 } \left( \frac { 4 t ^ { 2 } } { 5 R } \right)$
(4) $\cot ^ { - 1 } \left( \frac { R } { 20 t ^ { 2 } } \right)$

jee-main 2022 Q2 Finding Angle of Projection from Given Conditions View

A ball is projected from the ground with a speed $15 \mathrm{~m}\mathrm{~s}^{-1}$ at an angle $\theta$ with horizontal so that its range and maximum height are equal, then $\tan\theta$ will be equal to
(1) $\frac{1}{4}$
(2) $\frac{1}{2}$
(3) 2
(4) 4

jee-main 2022 Q2 Maximum Range or Maximum Height from Given Constraints View

A person can throw a ball upto a maximum range of 100 m. How high above the ground he can throw the same ball?
(1) 25 m
(2) 50 m
(3) 100 m
(4) 200 m

jee-main 2022 Q21 Range and Complementary Angle Relationships View

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 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.

jee-main 2023 Q3 Assertion-Reason on Projectile Concepts View

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 $45^{\circ}$, its 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:
(1) $A$ is false but $R$ is true
(2) $A$ is true but $R$ is false
(3) Both $A$ and $R$ are correct and $R$ is the correct explanation of $A$
(4) Both $A$ and $R$ are correct but $R$ is NOT the correct explanation of $A$

jee-main 2023 Q21 Determining Initial Conditions from Position or Flight Data View

A projectile fired at $30^\circ$ to the ground is observed to be at same height at time 3 s and 5 s after projection, during its flight. The speed of projection of the projectile is $\_\_\_\_$ $\mathrm{m}\mathrm{~s}^{-1}$. (Given $g = 10$ m s$^{-2}$)

jee-main 2024 Q2 Angular Momentum of a Projectile View

A particle of mass $m$ projected with a velocity $u$ making an angle of $30 ^ { \circ }$ with the horizontal. The magnitude of angular momentum of the projectile about the point of projection when the particle is at its maximum height $h$ is :
(1) $\frac { \sqrt { 3 } } { 16 } \frac { m u ^ { 3 } } { g }$
(2) $\frac { \sqrt { 3 } } { 2 } \frac { m u ^ { 2 } } { g }$
(3) $\frac { m u ^ { 3 } } { \sqrt { 2 } g }$
(4) zero

jee-main 2024 Q22 Angular Momentum of a Projectile View

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 $\dfrac{\sqrt{Z}\, m u^3}{X g}$. The value of $X$ is $\_\_\_\_$.

jee-main 2025 Q2 Finding Angle of Projection from Given Conditions View

Q2. The angle of projection for a projectile to have same horizontal range and maximum height is :
(1) $\tan ^ { - 1 } ( 4 )$
(2) $\tan ^ { - 1 } \left( \frac { 1 } { 4 } \right)$
(3) $\tan ^ { - 1 } \left( \frac { 1 } { 2 } \right)$
(4) $\tan ^ { - 1 } ( 2 )$

jee-main 2025 Q21 Ratio of Projectile Quantities for Different Launch Parameters View

Q21. The maximum height reached by a projectile is 64 m . If the initial velocity is halved, the new maximum height of the projectile is $\_\_\_\_$ m.

jee-main 2025 Q31 Range and Complementary Angle Relationships View

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:
(1) $\frac{1 - \tan\alpha}{1 + \tan\alpha}$
(2) $\frac{1 - \sin 2\alpha}{1 + \sin 2\alpha}$
(3) $\frac{1 + \sin 2\alpha}{1 - \sin 2\alpha}$
(4) $\frac{1 + \sin\alpha}{1 - \sin\alpha}$

jee-main 2025 Q50 Projectile from a Non-Inertial or Moving Frame View

The maximum speed of a boat in still water is $27\,\mathrm{km/h}$. Now this boat is moving downstream in a river flowing at $9\,\mathrm{km/h}$. A man in the boat throws a ball vertically upwards with speed of $10\,\mathrm{m/s}$. Range of the ball as observed by an observer at rest on the river bank, is \_\_\_\_ cm. (Take $g = 10\,\mathrm{m/s^2}$)

jee-main 2026 Q1 Velocity or Momentum at a Given Time View

A projectile is projected at angle of projection $60 ^ { \circ }$ with speed $u$. When its velocity makes an angle $45 ^ { \circ }$ with horizontal its speed is $20 \mathrm {~m} / \mathrm { s }$. Find u ?
(A) $\mathbf { 1 0 } \sqrt { \mathbf { 2 } }$
(B) $20 \mathrm {~m} / \mathrm { s }$
(C) $20 \sqrt { 2 } \mathrm {~m} / \mathrm { s }$
(D) $40 \mathrm {~m} / \mathrm { s }$

taiwan-gsat 2024 Q6 5 marks Multi-step composite figure problem View

On the same plane, two artillery batteries $A$ and $B$ are 7 kilometers apart, with $A$ directly east of $B$. During an exercise, $A$ fires a projectile west-northwest at angle $\theta$, and $B$ fires a projectile east-northwest at angle $\theta$, where $\theta$ is an acute angle. Both projectiles hit the same target $P$ 9 kilometers away. Then $A$ fires another projectile west-northwest at angle $\frac{\theta}{2}$, landing at point $Q$ 9 kilometers away. What is the distance $\overline{BQ}$ between artillery battery $B$ and landing point $Q$?
(1) 4 kilometers
(2) 4.5 kilometers
(3) 5 kilometers
(4) 5.5 kilometers
(5) 6 kilometers

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