Home
Class 12
PHYSICS
On one boundary of a swimming pool , the...

On one boundary of a swimming pool , there is a person at point `A` whose speed of running on group (boundary) is `10ms^(-1)` , while that of swimming is `6ms^(-1)` . He has to reach a point `B` in the swimming pool . The distance covered on the boundary so that the time required to reach the point `B` in the pool is minimum, is

A

`10`m

B

`6`m

C

`7`m

D

`sqrt(116)`m

Text Solution

Verified by Experts

The correct Answer is:
C

Required time `t= (x)/(10) + ((sqrt4^(2) + (10-x)^(2)))/(5)` for minimum time `(dt)/(dx) = 0 implies x=7"m"`
OR
Just like light , he has different speeds on ground and in water , so to minimize the time, Fermat's principle must hold good.
`"Sin"theta_(C) = (6)/(10) = (3)/(5) implies theta_(C) = 37^(@) implies "tan"theta_(C) = (3)/(4) = (10-x)/(4) implies x=7"m"`
Promotional Banner

Topper's Solved these Questions

  • GEOMETRICAL OPTICS

    ALLEN |Exercise EXERCISE -01|65 Videos

  • GEOMETRICAL OPTICS

    ALLEN |Exercise EXERCISE -02|82 Videos

  • GEOMETRICAL OPTICS

    ALLEN |Exercise subjective|14 Videos

  • CURRENT ELECTRICITY

    ALLEN |Exercise EX.II|66 Videos

  • GRAVITATION

    ALLEN |Exercise EXERCISE 4|9 Videos

Similar Questions

Explore conceptually related problems

An accelration produces a narrow beam of protons, each having an initial speed of v_(0) . The beam is directed towards an initially uncharges distant metal sphere of radius R and centered at point O. The initial path of the beam is parallel to the axis of the sphere at a distance of (R//2) from the axis, as indicated in the diagram. The protons in the beam that collide with the sphere will cause it to becomes charged. The subsequentpotential field at the accelerator due to the sphere can be neglected. The angular momentum of a particle is defined in a similar way to the moment of a force. It is defined as the moment of its linear momentum, linear replacing the force. We may assume the angular momentum of a proton about point O to be conserved. Assume the mass of the proton as m_(P) and the charge on it as e. Given that the potential of the sphere increases with time and eventually reaches a constant velue. One the potential of the sphere has reached its final, constant value, the minimum speed v of a proton along its trajectory path is given by

A man wishes to swim across a river 40 m wide flowing with a speed of 3m/s. such that he reaches the point just infront on the other bank in time not greater than 100s. The angle made by the direction he swims and river flow direction is :-

A swimmer starts to swim from point a to cross a river. He wants to reach point B on the opposite side of the river. The line AB makes an angle 60^(@) with the river flow as shown. The velocity of the swimmer in still water is same as that of the water (i) In what direction should he try to direct his velocity ? Calculate angle between his velocity and river velocity. (ii) Find the ratio of the time taken to cross the river in this situation to the minimum time in which he can cross this river.

A cricket ball is thrown at a speed of ms^(-1) in a direction 30^(@) above the horizontal . Calculate (a) the maximum height , (b) the time taken by the ball to return to the same thrower to the point where the ball returns to the same level.

The air column in a pipe closed at one end is made to vibrate in its second overtone by a tuning fork of frequency 440 Hz . The speed of sound in air is 330ms^(-1) . End corrections may be neglected. Let P_(0) denote the mean pressure at any point in the pipe, and DeltaP the maximum amplitude of pressure variation. (a) What the length L of the air column. (b) What is the amplitude of pressure variation at the middle of the column? ( c ) What are the maximum and minimum pressures at the open end of the pipe? (d) What are the maximum and minimum pressures at the closed end of the pipe?

A swimmer starts to swim from point a to cross a river. He wants to reach point B on the opposite side of the river. The line AB makes an angle 60^(@) with the river flow as hown. The velocity of the swimmer in still water is same as that of the water (i) In what direction should he try to direct his velocity ? Calculate angle between his velocity anf river velocity. (ii) Find the ratio of the time taken to cross the river in this situation to the minimum time in which he can cross this river.

A ray of light travelling with a speed c leaves point 1 shown in figure and is reflected to point 2. The ray strikes the reflecting surface at a distance x from point 1. According to Fermat's principle of least time, among all possible paths between two points , the one actually taken by a ray of light is that for which the time taken is the least (In fact there are some cases in which the time taken by a ray is maximum rather than a minimum). Find the time for the ray to reach from point 1 to point 2.

The brakes of a train which is travelling at 30 m//s are applied as the train passes point A. The brakes produce a constant retardation of magnitude 3lambda m//s^(2) until the speed of the train is reduced to 10 m//s . The train travels at this speed for a distance and is then uniformly accelerated at lambda m//s^(2) until it again reaches a speed of 30 m//s as it passes point B. the time taken by the train in travelling from A to B, a distance of 4 km, is 4 min. Sketch the speed time graph for this motion and calculate:- (i) The value of lambda (ii) Distance travelled at 10 m//s .

Thin films, including soap bubbles and oil show patterns of alternative dark and bright regions resulting from interference among the reflected light waves. If two waves are in phase, their crests and troughs will coincide. The interference will be constructive and the amplitude of resultant wave will be greater then either of constituent waves. If the two wave are not of phase by half a wavelength (180^(@)) , the crests of one wave will coincide width the troughs of the other wave. The interference will be destructive and the amplitude of the resultant wave will be less than that of either constiuent wave. 1. When incident light I, reaches the surface at point a, some of the light is reflected as ray R_(a) and some is refracted following the path ab to the back of the film. 2. At point b, some of the light is refracted out of the film and part is reflected back through the film along path bc. At point c, some of the light is reflected back into the film and part is reflected out of the film as ray R_(c) . R_(a) and R_(c) are parallel. However, R_(c) has traveled the extra distance within the film of abc. If the angle of incidence is small, then abc is approximately twice the film's thickness . If R_(a) and R_(c) are in phase, they will undergo constructive interference and the region ac will be bright. If R_(a) and R_(c) are out of phase, they will undergo destructive interference and the region ac will be dark. I. Refraction at an interface never changes the phase of the wave. II. For reflection at the interfere between two media 1 and 2, if n_(1) gt n_(2) , the reflected wave will change phase. If n_(1) lt n_(2) , the reflected wave will not undergo a phase change. For reference, n_(air) = 1.00 . III. If the waves are in phase after reflection at all intensities, then the effects of path length in the film are: Constrictive interference occurs when 2 t = m lambda // n, m = 0, 1,2,3 ,... Destructive interference occurs when 2 t = (m + (1)/(2)) (lambda)/(n) , m = 0, 1, 2, 3 ,... If the waves are 180^(@) out of the phase after reflection at all interference, then the effects of path length in the film area: Constructive interference occurs when 2 t = (m + (1)/(2)) (lambda)/(n), m = 0, 1, 2, 3 ,... Destructive interference occurs when 2 t = (m lambda)/(n) , m = 0, 1, 2, 3 ,... 72. A film with index of refraction 1.50 coats a glass lens with index of refraction 1.80. What is the minimum thickness of the thin film that will strongly reflect light with wavelength 600 nm?

Thin films, including soap bubbles and oil show patterns of alternative dark and bright regions resulting from interference among the reflected ligth waves. If two waves are in phase, their crests and troughs will coincide. The interference will be cosntructive and the amlitude of resultant wave will be greater then either of constituent waves. If the two wave are not of phase by half a wavelength (180^(@)) , the crests of one wave will coincide width the troughs of the other wave. The interference will be destructive and the ampliutde of the resultant wave will be less than that of either consituent wave. 1. When incident light I, reaches the surface at point a, some of the ligth is reflected as ray R_(a) and some is refracted following the path ab to the back of the film. 2. At point b, some of the light is refracted out of the film and part is reflected back through the film along path bc. At point c, some of the light is reflected back into the film and part is reflected out of the film as ray R_(c) . R_(a) and R_(c) are parallel. However, R_(c) has travelled the extra distance within the film fo abc. If the angle of incidence is small, then abc is approxmately twice the film's thickness . If R_(a) and R_(c) are in phase, they will undergo constructive interference and the region ac will be bright. If R_(a) and R_(c) are out of phase, they will undergo destructive interference and the region ac will be dark. I. Refraction at an interface never changes the phase of the wave. II. For reflection at the interfere between two media 1 and 2, if n_(1) gt n_(2) , the reflected wave will change phase. If n_(1) lt n_(2) , the reflected wave will not undergo a phase change. For reference, n_(air) = 1.00 . III. If the waves are in phase after reflection at all intensities, then the effects of path length in the film are: Constrictive interference occurs when 2 t = m lambda // n, m = 0, 1,2,3 ,... Destrcutive interference occurs when 2 t = (m + (1)/(2)) (lambda)/(n) , m = 0, 1, 2, 3 ,... If the waves are 180^(@) out of the phase after reflection at all interference, then the effects of path length in the film ara: Constructive interference occurs when 2 t = (m + (1)/(2)) (lambda)/(n), m = 0, 1, 2, 3 ,... Destructive interference occurs when 2 t = (m lambda)/(n) , m = 0, 1, 2, 3 ,... A thin film with index of refraction 1.33 coats a glass lens with index of refraction 1.50. Which of the following choices is the smallest film thickness that will not reflect light with wavelength 640 nm?

ALLEN -GEOMETRICAL OPTICS-SOME WORKED OUT EXAMPLES
  1. Choose the correct mirror image of

    Text Solution

    |

  2. A point object O can move along vertical line AB as shown in figure . ...

    Text Solution

    |

  3. On one boundary of a swimming pool , there is a person at point A whos...

    Text Solution

    |

  4. A person has D cm wide face and his two eyes are separated by d cm. Th...

    Text Solution

    |

  5. If x and y denote the distance of the object and image from the focus ...

    Text Solution

    |

  6. A concave mirror forms an image I corresponding to a point object O . ...

    Text Solution

    |

  7. A man of height 2 m stands on a straight road on a hot day. The vertic...

    Text Solution

    |

  8. A system of coordinatees is drawn in a medium whose refractive index v...

    Text Solution

    |

  9. A ray of light in incident along a vector hati + hatj - hatk on a plan...

    Text Solution

    |

  10. A ray of light is incident in situation as shown in figure. Which...

    Text Solution

    |

  11. A fish lies at the bottom of a 4m deep water lake. A bird flies 6m abo...

    Text Solution

    |

  12. A plane mirror and an object has speeds of 5 m//s "and" 10 m//s respec...

    Text Solution

    |

  13. A ray of light travelling with a speed c leaves point 1 shown in figur...

    Text Solution

    |

  14. A ray of light travelling with a speed c leaves point 1 shown in figur...

    Text Solution

    |

  15. A ray of light travelling with a speed c leaves point 1 shown in figur...

    Text Solution

    |

  16. One hard and stormy night you find yourself lost in the forest when yo...

    Text Solution

    |

  17. One hard and stormy night you find yourself lost in the forest when yo...

    Text Solution

    |

  18. One hard and stormy night you find yourself lost in the forest when yo...

    Text Solution

    |

  19. There is a spherical glass ball of refractive index mu(1) and another ...

    Text Solution

    |

  20. There is a spherical glass ball of refractive index mu(1) and another ...

    Text Solution

    |