Motion in a Plane

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JEE Main 2020, 6 Sep Shift-II

A particle moving in the xy-plane experiences a velocity dependent force $F = k (v_{y} \hat{i} + v_{x} \hat{j})$ , where $v_{x}$ and $v_{y}$ are the x and y components of its velocity $v$ . If $a$ is the acceleration of the particle, then which of the following statements is true for the particle?

v \times a

is constant in time.

(B)

F

arises due to a magnetic field.

(C)

Kinetic energy of particle is constant in time.

(D)

Quantity

v \cdot a

is constant in time.

JEE Main 2021, 16 March Shift-II

LEVELJEE Main

A mosquito is moving with a velocity $v = (0.5 t^{2} \hat{i} + 3 t \hat{j} + 9 \hat{k}) m/s$ and accelerating in uniform conditions. What will be the direction of mosquito after 2s?

(A)

tan^{- 1} (\frac{2}{3})

from X-axis

(B)

tan^{- 1} (\frac{2}{3})

from Y-axis

(C)

tan^{- 1} (\frac{5}{2})

from Y-axis

(D)

tan^{- 1} (\frac{5}{2})

from X-axis

JEE Main 2019

LEVELJEE Main

In three dimensional system, the position coordinates of a particle (in motion) are given below $x = a cos ω t, y = a sin ω t, z = aω t$ The velocity of particle will be

(A)

2 aω

(B)

2 aω

(C)

aω

(D)

3 aω

LEVELJEE Main

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

(A)

10 units

(B)

72

JEE Main 2020, 4 Sep Shift-I

LEVELJEE Main

Starting from the origin at time $t = 0$ , with initial velocity $5 \hat{j} ms^{- 1}$ , a particle moves in the xy-plane with a constant acceleration of $(10 \hat{i} + 4 \hat{j}) ms^{- 2}$ . At time $t$ , its coordinates are $(20 m, y_{0} m)$ . The values of $t$ and $y_{0}$ respectively, are

A particle is moving with velocity $v = k (y \hat{i} + x \hat{j})$ , where $k$ is a constant. The general equation for its path is

(A)

y = x^{2} + constant

(B)

y^{2} = x + constant

(C)

x y = constant

(D)

y^{2} = x^{2} + constant

LEVELJEE Main

The coordinates of a moving particle at any time $t$ are given by $x = α t^{3}$ and $y = β t^{3}$ . The speed of the particle at time $t$ is given by

(A)

3 t α^{2} + β^{2}

(B)

3 t^{2} α^{2} + β^{2}

(C)

t^{2} α^{2} + β^{2}

(D)

α^{2} + β^{2}

JEE Main 2019, 11 Jan Shift-II

LEVELJEE Main

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

(A)

5 m

(B)

202 m

(C)

102 m

(D)

15 m

LEVELJEE Main

A particle is moving Eastwards with a velocity of $5 ms^{- 1}$ . In $10 s$ , the velocity changes to $5 ms^{- 1}$ Northwards. The average acceleration in this time is

(A)

\frac{1}{2} ms^{- 2}

towards North-East

(B)

\frac{1}{2} ms^{- 2}

\frac{1}{2} ms^{- 2}

towards North-West

JEE Main 2019, 9 Jan Shift-I

LEVELJEE Main

A particle is moving with a velocity $v = k (y \hat{i} + x \hat{j})$ , where $k$ is a constant. The general equation for its path is

(A)

y = x^{2} + constant

(B)

y^{2} = x + constant

(C)

x y = constant

(D)

y^{2} = x^{2} + constant

JEE Main 2019, 9 April Shift-II

LEVELJEE Main

The position vector of particle changes with time according to the relation $r (t) = 15 t^{2} \hat{i} + (4 - 20 t^{2}) \hat{j}$ . What is the magnitude of the acceleration (in $ms^{- 2}$ ) at $t = 1$ ?

An object $A$ is kept fixed at the point $x = 3 m$ and $y = 1.25 m$ on a plank $P$ raised above the ground. At time $t = 0$ , the plank starts moving along the $+ x$ -direction with an acceleration $1.5 m/s^{2}$ . At the same instant, a stone is projected from the origin with a velocity $u$ as shown. A stationary person on the ground observes the stone hitting the object during its downward motion at an angle of $4 5^{\circ}$ to the horizontal. All the motions are in $x$ - $y$ plane. Find $u$ and the time after which the stone hits the object. (Take $g = 10 m/s^{2}$ ).

JEE Advanced 1999

LEVELJEE Advanced

The coordinates of a particle moving in a plane are given by $x (t) = a cos (pt)$ and $y (t) = b sin (pt)$ where $a, b$ ( $< a$ ) and $p$ are positive constants of appropriate dimensions. Then,

* Multiple Correct Options

(A)

the path of the particle is an ellipse

(B)

the velocity and acceleration of the particle are normal to each other at

t = \frac{π}{2 p}

(C)

the acceleration of the particle is always directed towards a focus

(D)

the distance travelled by the particle in time interval

t = 0

t = \frac{π}{2 p}

a

JEE Advanced 1999

LEVELBoard

In $1.0 s$ , a particle goes from point $A$ to point $B$ , moving in a semicircle (see figure). The magnitude of the average velocity is

(A)

3.14 m/s

(B)

2.0 m/s

(C)

1.0 m/s

A particle moves in a circle of radius $R$ . In half revolution its displacement is $\dots\dots$ and distance covered is $\dots\dots$ .

JEE Advanced 1987

LEVELJEE Main

Spotlight $S$ rotates in a horizontal plane with constant angular velocity of $0.1 rad/s$ . The spot of light $P$ moves along the wall at a distance of $3 m$ . The velocity of the spot $P$ when $θ = 4 5^{\circ}$ (see fig.) is……. $m/s$ .

JEE Advanced 1993

LEVELJEE Main

A particle $P$ is sliding down a frictionless hemispherical bowl. It passes the point $A$ at $t = 0$ . At this instant of time, the horizontal component of its velocity is $v$ . A bead $Q$ of the same mass as $P$ is ejected from $A$ at $t = 0$ along the horizontal string $A B$ , with the speed $v$ . Friction between the bead and the string may be neglected. Let $t_{P}$ and $t_{Q}$ be the respective times taken by $P$ and $Q$ to reach the point $B$ . Then

(A)

t_{P} < t_{Q}

(B)

t_{P} = t_{Q}

(C)

t_{P} > t_{Q}

(D)

\frac{t _{P}}{t _{Q}} = \frac{length of arc A C B}{length of chord A B}

Pathfinder for Olympiad and JEE Advanced Physics

LEVELJEE Advanced

At a particular instant of time, position vector $r$ , velocity vector $v$ and angular position $θ$ of a particle traversing a path AB are shown in the figure. Here $ϕ$ is the angle made by the velocity vector with the positive x-axis. Which of the following statements is/are correct?

* Multiple Correct Options

(A)

Modulus of angular velocity is

\frac{d θ}{d t} = \frac{v s i n ( ϕ - θ )}{r}

(B)

Modulus of tangential component of acceleration is

r \frac{d ^{2} θ}{d t ^{2}}

(C)

Modulus of normal component of acceleration is

v \frac{d θ}{d t}

(D)

Modulus of normal component of acceleration is

v \frac{d ϕ}{d t}

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?

Pathfinder for Olympiad and JEE Advanced Physics

LEVELJEE Advanced

A biker is moving with constant velocity $v$ away from a long straight wall at an angle $θ$ with the wall. He honks a short beep of horn when he is at a distance $l$ from the wall. After how long from the instant he has honked, will he again hear an echo of the honking? Speed of the sound in air is $c$ .

Pathfinder for Olympiad and JEE Advanced Physics

LEVELJEE Advanced

Two bikers simultaneously start a race with constant speeds from point A to traverse a triangular track ABC, one clockwise and the other in anticlockwise sense. They simultaneously cross at B first time after a time interval $Δ t_{1} = 4$ min. If they continue the race, how long after they cross at B first time will they again simultaneously cross at B?

Pathfinder for Olympiad and JEE Advanced Physics

LEVELJEE Advanced

Three ants A, B and C are crawling on a large horizontal tabletop always occupying vertices of an equilateral triangle, size of which may vary with time. If at an instant, speeds of A and B are $v_{A}$ and $v_{B}$ , which of the following conclusions can you make for speed $v_{C}$ of C?

(A)

v_{C} < 0.5 (v_{A} + v_{B})

(B)

v_{C} \leq 0.5 (v_{A} + v_{B})

(C)

v_{C} < v_{A} + v_{B}

(D)

v_{C} \leq v_{A} + v_{B}

Pathfinder for Olympiad and JEE Advanced Physics

LEVELJEE Advanced

A spacecraft is moving in space, where all the external forces can be neglected. Any change in its speed and direction of motion can be accomplished by rockets installed on it. At an instant when it is moving with a speed $v = 100 m/s$ , the crew inside decides to take a $9 0^{\circ}$ turn with an acceleration of constant modulus and then move in the new direction with the same speed $v$ . The rockets installed can provide a maximum acceleration $a = 52 m/s^{2}$ . Find the minimum time spent and shape of the path followed during the turn.

Pathfinder for Olympiad and JEE Advanced Physics

LEVELJEE Advanced

A horizontal wax bar B rests between a wedge and a vertical wall as shown in the figure. The wedge starts moving towards the wall with a constant acceleration $0.5 mm/s^{2}$ . The moment the wedge starts moving, a continuous supply of heat from the wall starts melting $1.0 mm$ length of the wax bar per second. If the bar always remains horizontal, which of the following conclusions can you make?

* Multiple Correct Options

(A)

The bar first moves downwards and then upwards.

(B)

The bar stops for a moment after

2.0 s

from the beginning.

(C)

Modulus of displacement of the bar in the first

4 s

1.2 mm

(D)

Distance travelled by the bar in the first

4 s

1.2 mm

Pathfinder for Olympiad and JEE Advanced Physics

LEVELJEE Advanced

Four identical rods are hinged at their ends to make a parallelogram ABCD. The hinged joint A is rigidly attached to a wall and the opposite joint C is pulled away from the wall with a constant acceleration $a$ as shown in the figure. Initially, the joints A and C were coincident. Find the acceleration vector of the joint B at the instant shown.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A mosquito is moving with a velocity v=(0.5t2i^+3tj^​+9k^) m/s and accelerating in uniform conditions. What will be the direction of mosquito after 2s?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } In three dimensional system, the position coordinates of a particle (in motion) are given below x=acosωt,y=asinωt,z=aωt The velocity of particle will be

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A particle has an initial velocity 3i^+4j^​ and an acceleration of 0.4i^+0.3j^​. Its speed after 10 s is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A particle is moving with velocity v=k(yi^+xj^​), where k is a constant. The general equation for its path is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The coordinates of a moving particle at any time t are given by x=αt3 and y=βt3. The speed of the particle at time t is given by

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A particle is moving Eastwards with a velocity of 5 ms−1. In 10 s, the velocity changes to 5 ms−1 Northwards. The average acceleration in this time is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A particle is moving with a velocity v=k(yi^+xj^​), where k is a constant. The general equation for its path is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The position vector of particle changes with time according to the relation r(t)=15t2i^+(4−20t2)j^​. What is the magnitude of the acceleration (in ms−2) at t=1?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The coordinates of a particle moving in a plane are given by x(t)=acos(pt) and y(t)=bsin(pt) where a,b (<a) and p are positive constants of appropriate dimensions. Then,

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } In 1.0 s, a particle goes from point A to point B, moving in a semicircle (see figure). The magnitude of the average velocity is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A particle moves in a circle of radius R. In half revolution its displacement is …… and distance covered is …… .

A mosquito is moving with a velocity $v = (0.5 t^{2} \hat{i} + 3 t \hat{j} + 9 \hat{k}) m/s$ and accelerating in uniform conditions. What will be the direction of mosquito after 2s?

In three dimensional system, the position coordinates of a particle (in motion) are given below $x = a cos ω t, y = a sin ω t, z = aω t$ The velocity of particle will be

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

A particle is moving with velocity $v = k (y \hat{i} + x \hat{j})$ , where $k$ is a constant. The general equation for its path is

The coordinates of a moving particle at any time $t$ are given by $x = α t^{3}$ and $y = β t^{3}$ . The speed of the particle at time $t$ is given by

A particle is moving Eastwards with a velocity of $5 ms^{- 1}$ . In $10 s$ , the velocity changes to $5 ms^{- 1}$ Northwards. The average acceleration in this time is

A particle is moving with a velocity $v = k (y \hat{i} + x \hat{j})$ , where $k$ is a constant. The general equation for its path is

The position vector of particle changes with time according to the relation $r (t) = 15 t^{2} \hat{i} + (4 - 20 t^{2}) \hat{j}$ . What is the magnitude of the acceleration (in $ms^{- 2}$ ) at $t = 1$ ?

The coordinates of a particle moving in a plane are given by $x (t) = a cos (pt)$ and $y (t) = b sin (pt)$ where $a, b$ ( $< a$ ) and $p$ are positive constants of appropriate dimensions. Then,

In $1.0 s$ , a particle goes from point $A$ to point $B$ , moving in a semicircle (see figure). The magnitude of the average velocity is

A particle moves in a circle of radius $R$ . In half revolution its displacement is $\dots\dots$ and distance covered is $\dots\dots$ .