A particle moves in x - y plane such that its position vector varies with time as r → = ( 2 sin 3 t ) i ^ + 2 1 - cos 3 t j ^ . Find the equation of the trajectory of the particle.

The given equation is, r → = 2 sin 3 t i ^ + 2 1 - cos 3 t j ^ Comparing above equation with the standard equation of r → = x i ^ + y j ^ , we get, x = 2 sin 3 t and y = 2 ( 1 - cos 3 t ) . x = 2 sin 3 t ⇒ sin 3 t = x 2 . . . 1 y = 2 ( 1 - cos 3 t ) . ⇒ cos 3 t = 1 - y 2 . . . . 2 Squaring & adding above two equations we get, sin 3 t 2 + cos 3 t 2 = x 2 4 + 1 - y 2 2 ⇒ x 2 4 + 1 - y 2 2 = 1

A particle moves in x - y plane such that its position vector varies with time as r → = ( 2 sin 3 t ) i ^ + 2 1 - cos 3 t j ^ . Find the equation of the trajectory of the particle.

The given equation is, r → = 2 sin 3 t i ^ + 2 1 - cos 3 t j ^ Comparing above equation with the standard equation of r → = x i ^ + y j ^ , we get, x = 2 sin 3 t and y = 2 ( 1 - cos 3 t ) . x = 2 sin 3 t ⇒ sin 3 t = x 2 . . . 1 y = 2 ( 1 - cos 3 t ) . ⇒ cos 3 t = 1 - y 2 . . . . 2 Squaring & adding above two equations we get, sin 3 t 2 + cos 3 t 2 = x 2 4 + 1 - y 2 2 ⇒ x 2 4 + 1 - y 2 2 = 1

A particle moves from position A to position B in a path as shown in figure. If the position vectors r → 1 and r → 2 making an angle θ between them are given, find the magnitude of displacement.

From the given diagram it is clear that the particle is moving from the point A t o B . We have to find out the magnitude of displacement of the particle, from triangle O A B : using triangle law of vector addition, O A → + A B → = O B → ⇒ r → 1 + Δ r → = r → 2 ⇒ r → 2 - r → 1 = Δ r → ⇒ s → = Δ r → = r → 2 - r → 1 Now the magnitude of the vector is given by, | s → | = | Δ r → | = r → 2 - r → 1 = r 1 2 + r 2 2 - 2 r 1 r 2 cosθ

EASY

JEE Advanced

IMPORTANT

Earn 100

A particle moves in $x - y$ plane such that its position vector varies with time as $\vec{r} = (2 \sin 3 t) \hat{i} + 2 (1 - \cos 3 t) \hat{j} .$ Find the equation of the trajectory of the particle.

Important Questions on Kinematics I

EASY

JEE Advanced

IMPORTANT

PHYSICS For Joint Entrance Examination JEE (Advanced) Mechanics I>Chapter 4 - Kinematics I>ILLUSTRATION 4.3

A particle moves from position $A$ to position $B$ in a path as shown in figure. If the position vectors ${\vec{r}}_{1}$ and ${\vec{r}}_{2}$ making an angle $θ$ between them are given, find the magnitude of displacement.

Question Image

EASY

JEE Advanced

IMPORTANT

PHYSICS For Joint Entrance Examination JEE (Advanced) Mechanics I>Chapter 4 - Kinematics I>ILLUSTRATION 4.4

A particle moves in a semicircular path of radius $R$ from $O$ to $A .$ Then it moves parallel to $z$ -axis covering a distance $R$ upto $B .$ Finally it moves along $B C$ parallel to $y$ -axis through a distance $2 R .$ Find the ratio of $\frac{D}{s} .$ (Here, $D$ is the distance covered and $s$ is the magnitude of the displacement.)

Question Image

MEDIUM

JEE Advanced

IMPORTANT

PHYSICS For Joint Entrance Examination JEE (Advanced) Mechanics I>Chapter 4 - Kinematics I>ILLUSTRATION 4.5>Q (a)

A particle is moving in a circle of radius $R$ (figure below).

$(a)$ What is its displacement when it covers $(i)$ half the circle, $(ii)$ full circle?

$(b)$ What is its distance when it covers $(i)$ half the circle and $(ii)$ full circle?

Question Image

MEDIUM

JEE Advanced

IMPORTANT

PHYSICS For Joint Entrance Examination JEE (Advanced) Mechanics I>Chapter 4 - Kinematics I>ILLUSTRATION 4.6

A train travels from city

A

to city

B

with a constant speed of

10 m s^{- 1}

and returns to the city

A

with a constant speed of

20 m s^{- 1}

. Find its average speed during its entire journey.

EASY

JEE Advanced

IMPORTANT

PHYSICS For Joint Entrance Examination JEE (Advanced) Mechanics I>Chapter 4 - Kinematics I>ILLUSTRATION 4.7

A particle starts from

x = 0

along a straight path, moves

25 m

straight, then stops. Then turns around and moves

10 m

back. Finally stops

15 m

from the start point. Total time of motion is

10 \sec

. What are his average speed and average velocity?

EASY

JEE Advanced

IMPORTANT

PHYSICS For Joint Entrance Examination JEE (Advanced) Mechanics I>Chapter 4 - Kinematics I>ILLUSTRATION 4.8

A car moving along a straight line moves with a constant velocity

v_{1}

for some time and with constant velocity

v_{2}

for the next equal time. What is the average velocity of the car?

EASY

JEE Advanced

IMPORTANT

PHYSICS For Joint Entrance Examination JEE (Advanced) Mechanics I>Chapter 4 - Kinematics I>ILLUSTRATION 4.9

A particle covers half its journey with a constant speed of

v

. Half the remaining part of the journey with a constant speed of

2 v

and the rest of the journey with a constant speed of

4 v

. Find its average speed during the entire journey.

EASY

JEE Advanced

IMPORTANT

PHYSICS For Joint Entrance Examination JEE (Advanced) Mechanics I>Chapter 4 - Kinematics I>ILLUSTRATION 4.10

A point traversed half the distance with a velocity

v_{0} .

The remaining part of the distance was covered with velocity

v_{1}

for half the time, and with velocity

v_{2}

for the other half of the time. Find the average speed of the point average over the whole time of motion.

A particle moves in x-y plane such that its position vector varies with time as r→=(2sin3t)i^+21-cos3tj^. Find the equation of the trajectory of the particle.

Important Questions on Kinematics I

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A particle moves in $x - y$ plane such that its position vector varies with time as $\vec{r} = (2 \sin 3 t) \hat{i} + 2 (1 - \cos 3 t) \hat{j} .$ Find the equation of the trajectory of the particle.