Relative Motion - Riverboat problem

Problem Statement:

A river has a width d = 40 m. The water flows at a constant velocity vA = – 6 i (m/s) with respect to a frame of reference at rest O . A boat wants to cross the river from point A on one bank to point B on the other (see figure). The velocity vector of the boat with respect to the water is: v’B = 4 j (m/s). A cyclist is crossing the bridge at a constant velocity vC = 2 j (m/s) with respect to O.

Relative Motion - Riverboat problem

Calculate:

  1. The velocity vector of the boat with respect to O.
  2. The velocity vector of the boat with respect to the cyclist.
  3. The time it takes for the boat to cross the river.
  4. The distance between points A and B.
  5. If the current had twice the speed, how long would it take for the boat to cross the river?

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Solution:

In this problem we have velocities measured with respect to two different frames of reference. The first one (O) is at rest and the second one, the water, which we will call O’, moves at constant velocity with respect to O. Since both are inertial, we will have to make use of the Galilean transformations.

More specifically, the Galilean transformation for the velocity is given by:

Where v is the velocity with respect to a frame of reference at rest and v’ is measured with respect to the frame of reference moving at constant velocity with respect to the first one.

Therefore in the equation above the velocity of the boat with respect to O will be v and¬†v’ will be the velocity of the boat with respect to the water (v’B in the problem statement). V is the velocity of the moving frame of reference with respect to the reference frame at rest (in this case the velocity of the water with respect to O, vA).

If we apply the Galilean transformation for the velocities to this problem we have:

And substituting the corresponding numerical values:


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In order to calculate the velocity of the boat with respect to the cyclist we use the Galilean transformation for the velocities, but in this case the frame of reference in motion O’ will be the cyclist. Therefore the Galilean transformation is now given by:

Substituting the givens, the velocity of the boat with respect to the cyclist is:


In order to calculate the time it takes for the boat to cross the river we use its width d and the vertical component of the velocity of the boat with respect to O. Since the last one is constant, the relationship between the space traveled by the boat in a straight line between the two banks, its speed on that axis and the time it takes is:


We can use this time to calculate the horizontal distance that the boat has traveled in the same amount of time. We use the horizontal component of its velocity:

And using the Pitagoras’ theorem we determine the distance between A and B:


If the current had twice the speed, it would take the same time for the boat to cross the river, since the vertical component of its velocity would be the same. However, it would travel a greater distance in the same amount of time.

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