math

A river flows at the speed of v from west to east. How should a man who is at a point A of the southern bank of the river, which is d meters away from the river, and wants to reach a point B on the northern bank of the river, which is 2d meters away from the river, choose his path in order to reach the destination at the shortest time? Assume the width of the river to be d, the man to run at a speed of 3v and swim at the speed of v/3 in still water, and the points A and B on the same meridian.

  1. 👍 0
  2. 👎 0
  3. 👁 455
  1. Let the man run to point P, which is x meters upstream from the line AB. He swims across, being swept y meters downstream, and then runs to B.

    We know that he is in the water for d/(v/3) = 3d/v seconds, so
    y = 3d/v * v = 3d.

    His path is thus

    √(d^2+x^2) meters on land,
    d meters in the water,
    √((2d)^2+(x-3d)^2) meters on land to B

    Now, dividing distance by speed, his time to travel is

    t = 1/(3v) (√(d^2+x^2)+√((2d)^2+(x-3d)^2)) + 3d/v
    =
    √(d^2+x^2)+√(4d^2+(x-3d)^2)+9d
    -----------------------------------------------
                           3v

    Now, to find minimum time, we need dt/dx = 0.

    Once you get dt/dx and simplify things a bit, you need to solve

    x√(d^2+x^2)+(x-3d)√(4d^2+(x-3d)^2) = 0

    I get x = 5d/3

    1. 👍 1
    2. 👎 1

Respond to this Question

First Name

Your Response

Similar Questions

  1. physics

    A river flows due east at 1.50 m/s. A boat crosses the river from the south shore to the north shore by maintaining a constant velocity of 9.2 m/s due north relative to the water. (a) What is the velocity of the boat as viewed by

  2. physics

    A 200-m-wide river flows due east at a uniform speed of 2.0 m/s. A boat with a speed of 8.0 m/s relative to the water leaves the south bank pointed in a direction west of north. What are the (a) magnitude and (b) direction of the

  3. Physics

    A boat starting on one side of the river heads to the south with a speed of 1.5 m/s, the river flows to the east at .8 m/s. What is the resultant velocity to the side of the river? And if the river is 50 m wide calculate the

  4. Physics

    A boat propelled so as to travel with a speed of 0.5m/s in still water, moves directly (in a straight line) across a river that is 60m wide. The river flows with a speed of 0.30m/s. How long in seconds does it take the boat to

  1. Physics

    A river flows toward the East at 1.5 m/s. Steve can swim 2.2 m/s in still water. If he wishes to cross the river so that his direction is northward, what direction relative to the water must he swim?

  2. Physic

    A river 1.00 mile wide flows with a constant speed of 1.00 mi/h. A woman leaves from a point on the river bank. The woman rows a boat 1.00 mi directly upstream and returns to the starting point. Her speed in still water is 2.00

  3. Physics

    A river flows from south to north at 5km/hr. On this river a boat is heading east to west perpendicular to the current at 7km/hr. As viewed by an eagle hovering at rest over the shore, how fast and in what direction is the boat

  4. Physics

    A river flows due south with a speed of 2.00 m/s. A man steers a motorboat across the river; his velocity relative to the water is 4.20 m/s due east. The river is 800 m wide. What is the magnitude of his velocity relative to the

  1. physics

    A boat propelled so as to travel with a speed of 0.50m/s in still water, moves directly (in a straight line) across a river that is 60m wide. The river flows with a speed of 0.30m/s. How long in seconds does it take the boat to

  2. PHYSICS

    A river is flowing from west to east at a speed of 8m/min, a man on the south bank of the river , capable of swimming at 20m/min in still water, wants to swim across the river in the shortest time. He should swim in a direction of

  3. Physics

    The captain of a boat wants to travel directly across a river that flows due east with a speed of 1.09 m/s. He starts from the south bank of the river and wants to reach the north bank by travelling straight across the river. The

  4. Physics

    Two ice skaters are practicing a lift. They skate directly towards each other. The woman has a mass of 60 kg and a velocity of 3 m/s towards the east. The man has a mass of 100 kg and is skating at a velocity of 4 m/s towards the

You can view more similar questions or ask a new question.