A particle moves so that its position vector is given by r → = c o s ω t x ^ + s i n ω t y ^ , where ω is a constant. Which of the following is true?

Velocity is perpendicular to r → and acceleration is directed towards the origin.

Velocity and acceleration both are perpendicular to r → .

Velocity and acceleration both are parallel to r → .

Velocity is perpendicular to r → and acceleration is directed away from the origin.

EASY

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A train starting from rest first accelerates uniformly up to speed $80 km h^{- 1}$ for time $t$ , then it moves with a constant speed for time $3 t$ . The average speed of the train for this duration of journey will be

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Important Questions on Motion in a Plane

EASY

Physics>Mechanics>Motion in a Plane>Distance and Displacement

A fly is in rectilinear motion as described by the graph. The average speed of the fly is

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MEDIUM

Physics>Mechanics>Motion in a Plane>Distance and Displacement

The displacement

x

(in meter) of a particle of mass

m

(in kg) moving in one dimension under the action of a force, is related to time

t

(in sec) by

t = \sqrt{x} + 3

. The displacement of the particle when its velocity is zero, will be

MEDIUM

Physics>Mechanics>Motion in a Plane>Distance and Displacement

The acceleration at the end of

2 s

, of a particle whose motion is represented by the equation

S = 4 t^{3} - 8 t^{2} + 5 t + 4

is ______

MEDIUM

Physics>Mechanics>Motion in a Plane>Distance and Displacement

Consider a particle moving along the positive direction of

X

-axis. The velocity of the particle is given by

v = α \sqrt{x}

(

α

is a positive constant). At time

t = 0

, if the particle is located at

x = 0

, the time dependence of the velocity and the acceleration of the particle are respectively.

MEDIUM

Physics>Mechanics>Motion in a Plane>Distance and Displacement

Consider an expanding sphere of instantaneous radius

R

whose total mass remains constant. The expansion is such that the instantaneous density

ρ

remains uniform throughout the volume. The rate of fractional change in density

(\frac{1}{ρ} \frac{d ρ}{d t})

is constant. The velocity

v

of any point on the surface of the expanding sphere is proportional to

EASY

Physics>Mechanics>Motion in a Plane>Distance and Displacement

A car moves in positive

Y

-direction with velocity

v

proportional to distance travelled

y

v (y) \propto y^{β}

, where

β

is a positive constant. The car covers a distance

L

with average velocity

<v>

proportional to

L

<v> \propto L^{1 / 3}

. The constant

β

is given as

MEDIUM

Physics>Mechanics>Motion in a Plane>Distance and Displacement

The displacement of a particle moving with uniform acceleration in time

t

is given by

S = 30 t + 5 t^{2}

, its initial velocity is _____

EASY

Physics>Mechanics>Motion in a Plane>Distance and Displacement

The displacement of a particle moving along a straight line is related to time as

\sqrt{x} = t - 3

. The displacement of the particle when its velocity is zero is

EASY

Physics>Mechanics>Motion in a Plane>Distance and Displacement

Average velocity of a particle executing SHM in one complete vibration is

MEDIUM

Physics>Mechanics>Motion in a Plane>Distance and Displacement

A particle is moving such that its position coordinates

(x, y)

are

(2 m, 3 m)

at time

t = 0,

(6 m, 7 m)

at time

t = 2 s

and

(13 m, 14 m)

at time

t = 5 s

. The average velocity vector

({\vec{V}}_{a v})

from

t = 0

t = 5 s

is:

EASY

Physics>Mechanics>Motion in a Plane>Distance and Displacement

If a particle's position is given by

x = 4 - 12 t + 3 t^{2}

where

t

is in the seconds and

x

in meters. What is its velocity at

t = 1 s

? Whether the particle is moving in positive

x

direction or negative

x

direction?

EASY

Physics>Mechanics>Motion in a Plane>Distance and Displacement

Two cars

P

and

Q

start from a point at the same time in a straight line and their positions are represented by

X_{P} (t) = a t + b t^{2}

and

X_{Q} (t) = f t - t^{2}

. At what time do the cars have the same velocity?

EASY

Physics>Mechanics>Motion in a Plane>Distance and Displacement

A bus travels at

110 km h^{- 1}

(kilometers per hour) on open highway. Its speed in metres per second is

MEDIUM

Physics>Mechanics>Motion in a Plane>Distance and Displacement

Consider a car initially at rest, starts to move along a straight road first with acceleration

5 m s^{- 2},

then with uniform velocity and finally, decelerating at

5 m s^{- 2},

, before coming to a stop. Total time taken from start to end is

t = 25 s

. If the average velocity during that time is

72 km {hr}^{- 1},

the car moved with uniform velocity for a time of

MEDIUM

Physics>Mechanics>Motion in a Plane>Distance and Displacement

A car covers the first half distance between two places at

40 kmph

and the other half at

60 kmph .

The average speed of the car is

EASY

Physics>Mechanics>Motion in a Plane>Distance and Displacement

A car moves from point A to point B at a speed of

40 km h^{- 1}

and returns at a speed of

60 km h^{- 1}

. Find its average velocity.

EASY

Physics>Mechanics>Motion in a Plane>Distance and Displacement

The motion of a particle is represented by the equation

S = 8 t^{3} - 2 t^{2} + 6 t + 7

. Find the velocity of the particle at the end of

2

seconds in

m / s

HARD

Physics>Mechanics>Motion in a Plane>Distance and Displacement

The position of a particle as a function of time

t,

is given by

x (t) = a t + b t^{2} - c t^{3}

where

a, b

and

c

are constants. When the particle has zero acceleration, then its velocity will be:

MEDIUM

Physics>Mechanics>Motion in a Plane>Distance and Displacement

The

x

and

y

coordinates of the particle at any time are

x = 5 t - 2 t^{2}

and

y = 10 t

respectively, where

x

and

y

are in meters and

t

is in seconds. The acceleration of the particle at

t = 2 s

MEDIUM

Physics>Mechanics>Motion in a Plane>Distance and Displacement

A particle moves so that its position vector is given by

\vec{r} = c o s ω t \hat{x} + s i n ω t \hat{y}

, where

ω

is a constant. Which of the following is true?

A train starting from rest first accelerates uniformly up to speed 80 km h-1 for time t, then it moves with a constant speed for time 3t. The average speed of the train for this duration of journey will be

Important Questions on Motion in a Plane

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A train starting from rest first accelerates uniformly up to speed $80 km h^{- 1}$ for time $t$ , then it moves with a constant speed for time $3 t$ . The average speed of the train for this duration of journey will be