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.

MEDIUM

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Each of the four particles moves along the $x$ -axis. Their coordinates (in meters) as function of time (in seconds) are given by

Particle $1 : x (t) = 3.5 - 2.7 t^{3}$

Particle $2 : x (t) = 3.5 + 2.7 t^{3}$

Particle 3: $x (t) = 3.5 + 2.7 t^{2}$

Particle 4: $x (t) = 2.5 - 3.4 t - 2.7 t^{2}$

which of these particles is speeding up for $t > 0$ ?

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Important Questions on Motion in a Straight Line

HARD

Physics>Mechanics>Motion in a Straight Line>Instantaneous Velocity and Speed

A stone thrown down with a seed

u

takes a time

t_{1}

to reach the ground, while another stone, thrown upwards from the same point with the same speed, takes time

t_{2}

. The maximum height the second stone reaches from the ground is

HARD

Physics>Mechanics>Motion in a Straight Line>Instantaneous Velocity and Speed

A parachutist with total weight

75 kg

drops vertically onto a sandy ground with a speed of

2 m s^{- 1}

and comes to a halt over a distance of

0.25 m

. The average force from the ground on her is close to-

EASY

Physics>Mechanics>Motion in a Straight Line>Instantaneous Velocity and Speed

A particle is moving along with

X

-axis such that its acceleration is proportional to the displacement from the equilibrium position and they are in the same direction. The displacement

x (t)

is given by (where

ω > 0

is a constant)

MEDIUM

Physics>Mechanics>Motion in a Straight Line>Instantaneous Velocity and Speed

A bullet loses

{(\frac{1}{n})}^{th}

of its velocity passing through one plank. Considering uniform retardation, the number of such planks that are required to stop the bullet can be:

EASY

Physics>Mechanics>Motion in a Straight Line>Instantaneous Velocity and Speed

In a car race on straight road, car

A

takes a time

t

less than car

B

at the finish and passes finishing point with a speed

v

more than that of car

B .

Both the cars start from rest and travel with constant acceleration

a_{1}

and

a_{2}

respectively. Then $v$ is equal to:

HARD

Physics>Mechanics>Motion in a Straight Line>Instantaneous Velocity and Speed

From a tower of height H, a particle is thrown vertically upwards with a speed u. The time taken by the particle, to hit the ground, is n times that taken by it to reach the highest point of its path.The relation between H, u and n is :

MEDIUM

Physics>Mechanics>Motion in a Straight Line>Instantaneous Velocity and Speed

Two skaters, $P$ and $Q$ are skating towards each other. Skater, $P$ throws a ball towards, $Q$ every $5 s$ such that it always leaves her hand with speed $2 m s^{- 1}$ with respect to the ground. Consider two cases: (I) $P$ runs with speed $1 m s^{- 1}$ towards $Q,$ while $Q$ remains stationary. (II) $Q$ runs with speed $1 m s^{- 1}$ towards $P,$ while $P$ remains stationary. Note That irrespective of speed of $P,$ ball always leaves $P$ s hand with speed $2 m s^{- 1}$ with respect to the ground. Ignore gravity. Balls will be received by $Q .$

EASY

Physics>Mechanics>Motion in a Straight Line>Instantaneous Velocity and Speed

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

EASY

Physics>Mechanics>Motion in a Straight Line>Instantaneous Velocity and Speed

A body moves along a straight line with acceleration

3 m s^{- 2}

for

2 s

and then with acceleration

4 m s^{- 2}

for

3 s.

What is his average acceleration?

EASY

Physics>Mechanics>Motion in a Straight Line>Instantaneous Velocity and Speed

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?

HARD

Physics>Mechanics>Motion in a Straight Line>Instantaneous Velocity and Speed

A particle of mass

m

is acted upon by a force

F

given by the empirical law

F = \frac{R}{t^{2}} v (t) .

If this law is to be tested experimentally by observing the motion starting from rest, the best way is to plot

HARD

Physics>Mechanics>Motion in a Straight Line>Instantaneous Velocity and Speed

A piece of wood of mass

0.03 kg

is dropped from the top of a

100 m

height building. At the same time, a bullet of mass

0.02 kg

is fired vertically upward, with a velocity

100 m s^{- 1},

from the ground. The bullet gets embedded in the wood. Then the maximum height to which the combined system reaches above the top of the building before falling below is:

(g = 10 m s^{- 2})

MEDIUM

Physics>Mechanics>Motion in a Straight Line>Instantaneous Velocity and Speed

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

MEDIUM

Physics>Mechanics>Motion in a Straight Line>Instantaneous Velocity and Speed

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 Straight Line>Instantaneous Velocity and Speed

A ball is dropped from the top of a

100 m

high tower on a planet. In the last

\frac{1}{2} s

before hitting the ground, it covers a distance of

19 m .

Acceleration due to gravity (in

m s^{- 2}

) near the surface on that planet is______

MEDIUM

Physics>Mechanics>Motion in a Straight Line>Instantaneous Velocity and Speed

The distance

x

covered by a particle in one dimensional motion varies with time

t

x^{2} = a t^{2} + 2 b t + c

. If the acceleration of the particle depends on

x

x^{- n}

, where

n

is an integer, the value of

n

is _________

EASY

Physics>Mechanics>Motion in a Straight Line>Instantaneous Velocity and Speed

A particle starts from origin $O$ from rest and moves with a uniform acceleration along the positive $x - a x i s$ . Identify all figures that correctly represent the motion qualitatively.
( $a =$ acceleration, $v =$ velocity, $x =$ dispalcement, $t =$ time)
$(A)$ Question Image
$(B)$
$(C)$
$(D)$

EASY

Physics>Mechanics>Motion in a Straight Line>Instantaneous Velocity and Speed

A body is thrown vertically upwards. Which one of the following graphs correctly represents the velocity

(v)

vs time

(t)

MEDIUM

Physics>Mechanics>Motion in a Straight Line>Instantaneous Velocity and Speed

In the cube of side $a$ shown in the figure, the vector from the central point of the face $A B O D$ to the central point of the face $B E F O$ will be:
Question Image

MEDIUM

Physics>Mechanics>Motion in a Straight Line>Instantaneous Velocity and Speed

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?

Each of the four particles moves along the x-axis. Their coordinates (in meters) as function of time (in seconds) are given by Particle 1:xt=3.5-2.7t3 Particle 2:xt=3.5+2.7t3 Particle 3: xt=3.5+2.7t2 Particle 4: xt=2.5-3.4t-2.7t2 which of these particles is speeding up for t>0?

Important Questions on Motion in a Straight Line

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