Core Syllabus Module

Kinematics

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Chapter Elite-20 Challenge Zone

Target Advanced

Pathfinder for Olympiad and JEE Advanced Physics

LEVELOlympiad

A particle P is moving with a constant speed $u$ on a straight line that makes an angle $θ$ with the positive $x$ -direction of a coordinate system. When P crosses the $y$ -axis at a point $(0, l)$ , another particle Q starts from the origin and chases P with a uniform speed $v$ ( $v > u$ ). The chaser Q always maintains its velocity vector towards the chased P. (a) How long after Q starts from the origin, will it catch P? (b) If both the chaser Q and the chased P move with equal speeds (i.e. $u = v$ ), what will be the minimum distance between them and what will be the maximum magnitude of acceleration of the chaser Q?

Pathfinder for Olympiad and JEE Advanced Physics

LEVELOlympiad

A stone projected from edge A of a high cliff strikes the ground at point C moving almost vertically. Reason for this strange behavior is air resistance that is proportional to the speed of the stone. The points A and B on the trajectory are in the same horizontal level. Time taken by the stone in its upward and downward motions above the level AB differ by $Δ t$ and moduli of vertical component of velocities at points A and B differ by $Δ v_{y}$ . Horizontal component of velocity at point A is $u_{x}$ and horizontal displacement of the stone from A to C is $R$ . Denoting acceleration due to gravity by $g$ , find suitable expression for the maximum height of the stone above the horizontal level AB.

JEE Advanced 1994

LEVELJEE Advanced

Two towers $A B$ and $C D$ are situated a distance $d$ apart as shown in figure. $A B$ is $20 m$ high and $C D$ is $30 m$ high from the ground. An object of mass $m$ is thrown from the top of $A B$ horizontally with a velocity of $10 m/s$ towards $C D$ . Simultaneously, another object of mass $2 m$ is thrown from the top of $C D$ at an angle of $6 0^{\circ}$ to the horizontal towards $A B$ with the same magnitude of initial velocity as that of the first object. The two objects move in the same vertical plane, collide in mid-air and stick to each other. (1994) (a) Calculate the distance $d$ between the towers. (b) Find the position where the objects hit the ground.

Pathfinder for Olympiad and JEE Advanced Physics

LEVELJEE Advanced

Two motorboats that can move with velocities 4.0 m/s and 6.0 m/s relative to water are going up-stream in a river. When the faster boat overtakes the slower boat, a buoy is dropped from the slower boat. After lapse of a time interval, both the boats turn back simultaneously and move at the same speeds relative to the water as before. Their engines are switched off when they reach the buoy again. If the maximum separation between the boats is 200 m after the buoy is dropped and water flow velocity in the river is 1.5 m/s, find distance between the places where the faster boat passes by the buoy.

Pathfinder for Olympiad and JEE Advanced Physics

LEVELJEE Advanced

Starting from the centre of a circular path of radius $R$ , a particle P chases another particle Q that is moving with a uniform speed $v$ on the circular path. The chaser P moves with a constant speed $u$ and always remains collinear with the centre and the location of the chased Q. (a) On which path will P eventually move and how long will it take to reach on this path? Consider the cases $u < v$ , $u = v$ and $u > v$ . (b) If speeds of the particles are $v = 4 m/s$ and $u = 8 m/s$ and radius of the circular path is $R = 84 m$ , how long P will take to reach Q. Use $π = 22/7$ .

Pathfinder for Olympiad and JEE Advanced Physics

LEVELJEE Advanced

A boy starts from point A and passes point C of a track ABC shown in the figure. Portion AB of length $l$ is straight and portion BC is a semicircle of radius $r$ ( $r < l$ ). Anywhere on the track, the modulus of the maximum acceleration of the boy is $a$ . Find minimum transit time of the boy from A to C.

Pathfinder for Olympiad and JEE Advanced Physics

LEVELJEE Advanced

Two balls A and B are simultaneously released on two frictionless inclined planes from the positions shown. The inclined planes have equal inclinations. The balls pass through a particular horizontal level 12 s and 4 s after they were released. How long after they were released will they be closest to each other?

Pathfinder for Olympiad and JEE Advanced Physics

LEVELJEE Advanced

A cylindrical pipe of radius $r$ is rolling towards a frog sitting on the horizontal ground. Centre of the pipe is moving with a constant velocity $v$ . To save itself, the frog jumps off and passes over the pipe touching it only at the top. Denoting air-time of the frog by $T$ , horizontal range of the jump by $R$ and acceleration due to gravity by $g$ , which of the following conclusions can you make?

* Multiple Correct Options

(A)

T = 4 \frac{r}{g}

(B)

T \geq 4 \frac{r}{g}

(C)

R = 4 (g r - v) \frac{r}{g}

(D)

R \geq 4 (g r - v) \frac{r}{g}

Pathfinder for Olympiad and JEE Advanced Physics

LEVELJEE Advanced

A fun drive in an amusement park runs between two spots that are $2.0 km$ apart. For safety reasons, acceleration of the drive is limited to $\pm 4.0 m/s^{2}$ , and the jerk i.e. rate of change in acceleration, is limited to $\pm 1.0 m/s^{3}$ . If the drive can achieve a maximum speed of $144 km/h$ , find the shortest transit time of the drive between the spots.

Pathfinder for Olympiad and JEE Advanced Physics

LEVELJEE Advanced

A honeybee is flying parallel to a tabletop at a height $h = 2.0$ m with a constant velocity $v = 202$ m/s. With its wings, it can achieve a maximum acceleration $a = 4003$ m/s $^{2}$ . At an instant when the honeybee is vertically above a honey drop on the tabletop, it decides to reach the honey drop. Neglect the reaction time of the honeybee and find the minimum time in which the honeybee can reach the honey drop.

Pathfinder for Olympiad and JEE Advanced Physics

LEVELJEE Advanced

Three particles A, B and C start moving simultaneously with constant velocities from three places. The starting places are collinear and that of B is somewhere in between those of A and C. In the absence of C, particles A and B would have collided $t_{1}$ time after they started and in the absence of B, particles A and C would have collided $t_{2}$ time after they started. What would have happened, if A were not present?

* Multiple Correct Options

(A)

t_{2} > t_{1}

, particles B and C may or may not collide.

(B)

t_{2} > t_{1}

, particles B and C collide in the interval

(t_{1}, t_{2})

(C)

t_{2} \leq t_{1}

, particles B and C must collide at an instant

t \leq t_{2}

(D)

t_{2} \leq t_{1}

, particles B and C must collide in the interval

[t_{1}, t_{2}]

Pathfinder for Olympiad and JEE Advanced Physics

LEVELJEE Advanced

When you start your stopwatch, a particle moving on the $x$ -axis is observed somewhere between the positions $x = 10$ m and $x = 12$ m. Sometime during the fourth second, it passes the position $x = 22$ m and at the instant $t = 12$ s it is observed somewhere between the positions $x = 55$ m and $x = 60$ m. When do you expect its arrival at the position $x = 88$ m?

Pathfinder for Olympiad and JEE Advanced Physics

LEVELJEE Advanced

One end of a light inextensible thread of length $l$ is held stationary over a frictionless horizontal floor while a small bead tied at the other end of the thread is describing a circular path with a uniform speed $v_{0}$ on the floor as shown in the figure. The upper end of the thread is suddenly pulled vertically upwards with a constant acceleration $a_{0}$ . If the bead does not leave the floor, find magnitude of its acceleration immediately after the upper end of the thread is pulled.

Pathfinder for Olympiad and JEE Advanced Physics

LEVELJEE Advanced

Two cars are moving at constant speeds; one on a circular path of radius $R = 200 m$ and the other on a straight road. Magnitude $v$ of velocity of one car relative to the other has been recorded at regular intervals of time and data thus obtained is represented in a graph as shown in the figure. Calculate speeds of both the cars relative to the ground.

Pathfinder for Olympiad and JEE Advanced Physics

LEVELJEE Advanced

Two boys enter a running escalator at the ground floor of a shopping mall. The first boy repeatedly follows a cycle of $p_{1} = 1$ step up and then $q_{1} = 2$ steps down whereas the second boy repeatedly follows a cycle of $p_{2} = 2$ steps up and then $q_{2} = 1$ step down. Both of them move relative to escalator with a speed $v_{r} = 50 cm/s$ . If the boys take $t_{1} = 250 s$ and $t_{2} = 50 s$ respectively to reach the first floor in complete numbers of cycles, how fast is the escalator running?

Pathfinder for Olympiad and JEE Advanced Physics

LEVELJEE Advanced

Two boys are standing near the ends of a 100 m long conveyor belt that is running with a constant velocity 1.0 m/s. The boys step on the conveyer belt on its opposite ends and start walking towards each other. After meeting, they immediately return towards the ends of the belt and then continue the process repeatedly. The boys walk with a constant speed of 3 m/s relative to the conveyer belt. What distance relative to the ground do the boys walk in the first 300 s?

Pathfinder for Olympiad and JEE Advanced Physics

LEVELJEE Advanced

In a battle field of ancient times, a soldier with a catapult stationed on the top of a very high cliff notices camps of enemy close to the bottom of the cliff as shown in the figure. Stones can be launched from the catapult with a speed $u = 40 m/s$ at an angle $θ = 6 0^{\circ}$ above the horizontal. If the air resistance reduces speed of the stone at a rate $k = 0.1 (m/s)/m$ and there is no wind, at what horizontal separation from the enemy camps should the soldier install the catapult to hit the enemy camps?

Pathfinder for Olympiad and JEE Advanced Physics

LEVELJEE Advanced

A stick AB of length $l$ stands vertically on a horizontal floor leaning on a wall. A beetle P starts climbing the stick from the floor. When the beetle starts climbing, the lower end B of the stick is made to move away from the wall with a constant velocity $v$ . The beetle climbs the stick with a constant speed $u$ relative to the stick. If the upper end A does not leave the wall, what maximum height can the beetle rise?

Pathfinder for Olympiad and JEE Advanced Physics

LEVELJEE Advanced

Comprehension Passage

Two particle A and B are moving towards each other on a straight line with equal speeds

5 m/s

. At an instant that is assumed

t = 0 s

, distance between the particles is

100 m

. It is desired to move another particle C always maintaining a distance

40 m

from the particle A and

30 m

from the particle B.

Question 1:

When and for how long can the particle C fulfil the given condition?

* Multiple Correct Options

(A)

3 s \leq t \leq 9 s

(B)

9 s \leq t \leq 11 s

(C)

11 s \leq t \leq 17 s

(D)

3 s \leq t \leq 17 s

Question 2:

What is speed of the particle C at the instant $t = 5 s$ ?

* Multiple Correct Options

(A)

3.5 m/s

(B)

4.8 m/s

(C)

5 m/s

(D)

6 m/s

Question 3:

What is modulus of acceleration of the particle C at the instant $t = 5 s$ ?

* Multiple Correct Options

(A)

2.3 m/s^{2}

(B)

0.58 m/s^{2}

(C)

0.90 m/s^{2}

(D)

3.30 m/s^{2}

Question 4:

At the instant, when the line joining locations of A and B is perpendicular to the line joining locations of B and C, what are the magnitudes of velocities of C relative to A and B respectively?

* Multiple Correct Options

(A)

0 m/s

and

5 m/s

(B)

0 m/s

and

10 m/s

(C)

3.75 m/s

and

0 m/s

(D)

3.75 m/s

and

10 m/s

Pathfinder for Olympiad and JEE Advanced Physics

LEVELJEE Advanced

A marble is projected in a viscous fluid, with an initial speed $u$ at an angle $θ$ above the horizontal. Drag force of the fluid results in an acceleration $a_{D} = - k v$ in addition to that of gravity, where $k$ is a positive constant and $v$ is the velocity of marble. Determine position coordinates and $x$ (horizontal) and $y$ (vertical) components of velocity of the marble as functions of time $t$ . What is the terminal velocity?

Micro-Topic Strategy Matrix

Motion in a Straight Line

Motion Graphs

Equations of Kinematics

Vector Addition, Subtraction, and Resolution

Dot and Cross Products

Relative Velocity

Motion in a Plane

Projectile Motion

Kinematics of Circular Motion

Chapter Elite-20 Challenge Zone

Comprehension Passage

When and for how long can the particle C fulfil the given condition?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } What is speed of the particle C at the instant t=5 s?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } What is modulus of acceleration of the particle C at the instant t=5 s?

At the instant, when the line joining locations of A and B is perpendicular to the line joining locations of B and C, what are the magnitudes of velocities of C relative to A and B respectively?

What is speed of the particle C at the instant $t = 5 s$ ?

What is modulus of acceleration of the particle C at the instant $t = 5 s$ ?