Two particles move in a uniform gravitational field with an acceleration g . At the initial moment the particles were located at one point and moved with velocities v 1 → = 3 . 0 m s - 1 and v 2 → = 4 . 0 m s - 1 horizontally in opposite directions. Find the distance between the particles at the moment when their velocity vectors become mutually perpendicular. (Take g = 9 . 8 m s - 2 )

Let the velocities of the particles (say v 1 ' → and v 2 ' → ) becomes mutually perpendicular after a time t . So, v 1 ' → = v 1 → + g → t , v 1 ' → = v 2 → + g → t . . . 1 As v 1 ' → ⊥ v 2 ' → so, v 1 ' → · v 2 ' → = 0 or v 1 → + g → t · v 2 → + g → t = 0 or - v 1 v 2 + g 2 t 2 = 0 , (As v → 1 and v → 2 are opposite to each other) Hence, t = v 1 v 2 g . . . 3 Since both the particles start from the same point, their initial relative position r → 0 12 = r → 0 1 - r → 0 2 = 0 . Both the particles moves under Earth's gravitational field with same acceleration due to gravity. Therefore, their relative acceleration, w → 12 = g → - g → = 0 . Their initial relative velocity, v → 0 12 = v → 1 + v → 2 = v 1 i ^ - - v 2 i ^ = v 1 + v 2 i ^ Now, from the second equation of uniformly accelerated motion, r → 12 = r → 0 12 + v → 0 12 t + 1 2 w 12 → t 2 We get, r → 12 = v → 0 12 t , ( Here, w → 12 = 0 and r → 0 ( 12 ) = 0 Hence, the sought distance is, r → 12 = v 1 + v 2 g v 1 v 2 , as v → 0 ( 12 ) = v 1 + v 2 r → 12 = 3 + 4 9 . 8 3 × 4 = 2 . 47 m

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Two boats, $A$ and $B$ , move away from a buoy anchored at the middle of a river along the mutually perpendicular straight lines- the boat $A$ along the river, and the boat $B$ across the river. Having moved off an equal distance from the buoy the boats returned. Find the ratio of times of motion of boats $\frac{t_{A}}{t_{B}}$ , if the velocity of each boat with respect to water is $η = 1.2$ times greater than the stream velocity.

Important Questions on PHYSICAL FUNDAMENTALS OF MECHANICS

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JEE Main

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Problems in General Physics>Chapter 1 - PHYSICAL FUNDAMENTALS OF MECHANICS>KINEMATICS>Q 1.9.

A boat moves relative to water with a velocity which is

η = 2.0

times less than the river flow velocity. At what angle to the stream direction must the boat move to minimize drifting?

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IMPORTANT

Problems in General Physics>Chapter 1 - PHYSICAL FUNDAMENTALS OF MECHANICS>KINEMATICS>Q 1.10.

Two bodies were thrown simultaneously from the same point -one, straight up, and the other, at an angle of

θ = 60 °

to the horizontal. The initial velocity of each body is equal to

\frac{t_{A}}{t_{B}} = 1.8

. Neglecting the air drag, find the distance between the bodies

t = 1.70

s

later.

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JEE Main

IMPORTANT

Problems in General Physics>Chapter 1 - PHYSICAL FUNDAMENTALS OF MECHANICS>KINEMATICS>Q 1.11.

Two particles move in a uniform gravitational field with an acceleration

g

. At the initial moment the particles were located at one point and moved with velocities

\vec{v_{1}} = 3.0 m s^{- 1}

and

\vec{v_{2}} = 4.0 m s^{- 1}

horizontally in opposite directions. Find the distance between the particles at the moment when their velocity vectors become mutually perpendicular. (Take

g = 9.8 m s^{- 2}

)

HARD

JEE Main

IMPORTANT

Problems in General Physics>Chapter 1 - PHYSICAL FUNDAMENTALS OF MECHANICS>KINEMATICS>Q 1.12.

Three points are located at the vertices of an equilateral triangle whose side equals

a

. They all start moving simultaneously with velocity

v

constant in modulus, with the first point heading continually for the second, the second for the third, and the third for the first. How soon will the points converge?

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JEE Main

IMPORTANT

Problems in General Physics>Chapter 1 - PHYSICAL FUNDAMENTALS OF MECHANICS>KINEMATICS>Q 1.13.

Point

A

moves uniformly with velocity

\vec{v}

so that the vector

\vec{v}

is continually aimed at point

B

which in its turn moves rectilinearly and uniformly with velocity

u < v

. At the initial moment of time

v ⊥ u

and the points are separated by a distance

l

. How soon will the points converge?

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JEE Main

IMPORTANT

Problems in General Physics>Chapter 1 - PHYSICAL FUNDAMENTALS OF MECHANICS>KINEMATICS>Q 1.14.

A train of length

l = 350 m

starts moving rectilinearly with constant acceleration

w = 3.0 \times 10^{- 2} m s^{- 2}, t = 30 s

after the start the locomotive headlight is switched on (event

1

), and

τ = 60 s

after that event the tail signal light is switched on (event

2

). Find the distance between these events in the reference frames fixed to the train and to the earth. How and at what constant velocity

V

relative to the earth must a certain reference frame

K

move for the two events to occur in it at the same point?

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JEE Main

IMPORTANT

Problems in General Physics>Chapter 1 - PHYSICAL FUNDAMENTALS OF MECHANICS>KINEMATICS>Q 1.15.

An elevator car whose floor to ceiling distance is equal to $2.7 m$ , starts ascending with constant acceleration $1.2 m s^{- 2}, 2.0 s$ , after the start a bolt begins falling from the ceiling of the car. Find,
$(a)$ the bolt's free-fall time,
$(b)$ the displacement and the distance covered by the bolt during the free-fall in the reference frame fixed to the elevator shaft.

(Take $g = 9.8 {ms}^{- 2}$ )

HARD

JEE Main

IMPORTANT

Problems in General Physics>Chapter 1 - PHYSICAL FUNDAMENTALS OF MECHANICS>KINEMATICS>Q 1.16.

Two particles,

1

and

2

, move with constant velocities

v_{1}

and

v_{2}

along two mutually perpendicular straight lines toward the intersection point

O

. At the moment

t = 0

the particles were located at the distances

l_{1}

and

l_{2}

from the point

O

. How soon will the distance between the particles become the smallest? What is it equal to?

Important Questions on PHYSICAL FUNDAMENTALS OF MECHANICS

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