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A battalion of soldiers is ordered to swim across a river $500 m$ wide. At what minimum rate should swim perpendicular to river flow in order to avoid being washed away by the waterfall $300 m$ down shore? The speed of current being $3 m s^{- 1}$

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Important Questions on Motion in a Plane

MEDIUM

Physics>Mechanics>Motion in a Plane>Relative Velocity

The least value of

α \in R

for which,

4 α x^{2} + \frac{1}{x} \geq 1,

for all

x > 0,

MEDIUM

Physics>Mechanics>Motion in a Plane>Relative Velocity

A river has a steady speed of

v

. A man swims upstream at a distance of

d

and swims back to the starting point in total time

t

. The man can swim at a speed of

2 v

in still water. If the time taken by the man in still water is

t_{0}

to complete the same length of swim, then

\frac{t}{t_{0}}

HARD

Physics>Mechanics>Motion in a Plane>Relative Velocity

Let

f : R \to (0, \infty)

and

g : R \to R

be twice differentiable functions such that

f^{″}

and

g^{″}

are continuous functions on

R

. Suppose

f^{'} (2) = g (2) = 0, f^{″} (2) \neq 0

and

g^{'} (2) \neq 0

\lim_{x \to 2} \frac{f (x) g (x)}{f^{'} (x) g^{'} (x)} = 1

, then

HARD

Physics>Mechanics>Motion in a Plane>Relative Velocity

The maximum value of

f (x) = \frac{x}{4 + x + x^{2}}

[- 1, 1]

MEDIUM

Physics>Mechanics>Motion in a Plane>Relative Velocity

A boat is moving from the east bank to the west bank on a south flowing river. If the speed of the boat is

4 km / h

and that of the river is

3 km / h .

If the width of the river is

2 km,

the distance travelled by the boat is

HARD

Physics>Mechanics>Motion in a Plane>Relative Velocity

If the function

f

given by

f (x) = x^{3} - 3 (a - 2) x^{2} + 3 a x + 7

, for some

a \in R

is increasing in

(0, 1]

and decreasing in

[1, 5)

, then a root of the equation,

\frac{f (x) - 14}{{(x - 1)}^{2}} = 0, (x \neq 1)

is :

MEDIUM

Physics>Mechanics>Motion in a Plane>Relative Velocity

The speed of a swimmer is

4 km h^{- 1}

in still water. If the swimmer makes his strokes normal to the flow of river of width

1 km

, he reaches a point

750 m

down the stream on the opposite bank. The speed of the river water is ______

km h^{- 1}

HARD

Physics>Mechanics>Motion in a Plane>Relative Velocity

A solid hemisphere is mounted on a solid cylinder, both having equal radii. If the whole solid is to have a fixed surface area and the maximum possible volume, then the ratio of the height of the cylinder to the common radius is

HARD

Physics>Mechanics>Motion in a Plane>Relative Velocity

A swimmer wants to cross a river from point $A$ to point $B$ . Line AB makes an angle of $30 °$ with the flow of the river. The magnitude of the velocity of the swimmer is the same as that of the river. The angle $θ$ with the line $A B$ should be _______ $°$ , so that the swimmer reaches point $B$ .

Question Image

HARD

Physics>Mechanics>Motion in a Plane>Relative Velocity

A motorboat whose speed is

15 km h^{- 1}

in still water goes

30 km

downstream and comes back in a total of

4

hours

30

minutes. What is the speed of the stream (in

km h^{- 1}

MEDIUM

Physics>Mechanics>Motion in a Plane>Relative Velocity

From the top of a

64

metres high tower, a stone is thrown upwards vertically with the velocity of

48 m / s .

The greatest height (in metres) attained by the stone, assuming the value of the gravitational acceleration

g = 32 m / s^{2}

, is:

MEDIUM

Physics>Mechanics>Motion in a Plane>Relative Velocity

A ball is dropped from a bridge that is

45 m

above the water. It falls directly into a boat which is moving with constant velocity. The boat is

12 m

away from the point of impact when the ball is dropped. The speed of the boat is (Take

g = 10 m s^{- 2}

)

HARD

Physics>Mechanics>Motion in a Plane>Relative Velocity

Let

k

and

K

be the minimum and the maximum values of the function

f (x) = \frac{{(1 + x)}^{0.6}}{1 + x^{0.6}} in [0, 1]

, respectively, then the ordered pair

(k, K)

is equal to:

HARD

Physics>Mechanics>Motion in a Plane>Relative Velocity

The maximum area (in sq. units) of a rectangle having its base on the

x -

axis and its other two vertices on the parabola,

y = 12 - x^{2}

such that the rectangle lies inside the parabola, is :

MEDIUM

Physics>Mechanics>Motion in a Plane>Relative Velocity

x = - 1

and

x = 2

are extreme points of

f (x) = α \log |x| + β x^{2} + x

, then

MEDIUM

Physics>Mechanics>Motion in a Plane>Relative Velocity

Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is:

HARD

Physics>Mechanics>Motion in a Plane>Relative Velocity

The maximum area of a rectangle that can be inscribed in a circle of radius

2

units is

EASY

Physics>Mechanics>Motion in a Plane>Relative Velocity

A person can row a boat at

10 km h^{- 1}

in still water. He takes two and half hours to row from A to B and back. If the distance between A and B is

12 km

, then the speed of the stream is :

EASY

Physics>Mechanics>Motion in a Plane>Relative Velocity

A person is swimming with a speed of

10 m s^{- 1}

at an angle of

120 °

with the flow and reaches to a point directly opposite on the other side of the river. The speed of the flow is

x m s^{- 1} .

The value of

x

to the nearest integer is ______.

EASY

Physics>Mechanics>Motion in a Plane>Relative Velocity

The stream of a river is flowing with a speed of

2 k m h^{- 1}

. A swimmer can swim at a speed of

4 km h^{- 1}

. The direction of the swimmer with respect to the flow of the river, to cross the river straight, is

A battalion of soldiers is ordered to swim across a river 500 m wide. At what minimum rate should swim perpendicular to river flow in order to avoid being washed away by the waterfall 300 m down shore? The speed of current being 3 m s-1

Important Questions on Motion in a Plane

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A battalion of soldiers is ordered to swim across a river $500 m$ wide. At what minimum rate should swim perpendicular to river flow in order to avoid being washed away by the waterfall $300 m$ down shore? The speed of current being $3 m s^{- 1}$