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A man wishes to cross a river in a boat. If he crosses the river in minimum time, he takes $10$ minutes with a drift of $120 m$ . If he crosses the river taking the shortest route, he takes $12.5$ minutes. Find the velocity of the boat with respect to water.

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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 man wishes to cross a river in a boat. If he crosses the river in minimum time, he takes 10 minutes with a drift of 120 m. If he crosses the river taking the shortest route, he takes 12.5 minutes. Find the velocity of the boat with respect to water.

Important Questions on Motion in a Plane

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A man wishes to cross a river in a boat. If he crosses the river in minimum time, he takes $10$ minutes with a drift of $120 m$ . If he crosses the river taking the shortest route, he takes $12.5$ minutes. Find the velocity of the boat with respect to water.