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Solution: The boat crosses the river by the shortest path if it moves perpendicular to the river current. The velocity of the boat in still water is equal to the relative velocity of the boat w. Question: You cannot cross a river to an exactly opposite point if your speed in still water is less than the speed of river water.

True B. He should swim in a direction? The velocity of the man in still water is equal to the relative velocity of the man w. Hence, the man takes the shortest time when he swims perpendicular to the river velocity i. Problem: A boat must get from point A to point B on the opposite bank of the river moving along a straight line AB that makes degree angle with the flow direction.

If distance AB is 2. The velocity of the boat relative to the ground should be along the line AB. Question: To cross a river in minimum time, your speed relative to the water should be perpendicular to the river flow. At what angle to the stream direction must the boat move to minimize drifting? Solution: The drift is zero if boat reaches directly opposite point. The boat cannot cross the river to an exactly opposite point. Exercise: A boat must get from point A to point B on the opposite bank of the river.

The distances BC and AC are equal. What is the minimum speed of the boat relative to the water? Exercise: A man can swim with a speed of 4. For example, an airplane usually encounters a wind - air that is moving with respect to an observer on the ground below.

As another example, a motorboat in a river is moving amidst a river current - water that is moving with respect to an observer on dry land. In such instances as this, the magnitude of the velocity of the moving object whether it be a plane or a motorboat with respect to the observer on land will not be the same as the speedometer reading of the vehicle.

Motion is relative to the observer. The observer on land, often named or misnamed the "stationary observer" would measure the speed to be different than that of the person on the boat.

The observed speed of the boat must always be described relative to who the observer is. To illustrate this principle, consider a plane flying amidst a tailwind. A tailwind is merely a wind that approaches the plane from behind, thus increasing its resulting velocity. The resultant velocity of the plane that is, the result of the wind velocity contributing to the velocity due to the plane's motor is the vector sum of the velocity of the plane and the velocity of the wind.

This resultant velocity is quite easily determined if the wind approaches the plane directly from behind. Since a headwind is a wind that approaches the plane from the front, such a wind would decrease the plane's resulting velocity.

This is depicted in the diagram below. Now what would the resulting velocity of the plane be? This question can be answered in the same manner as the previous questions. The resulting velocity of the plane is the vector sum of the two individual velocities. To determine the resultant velocity, the plane velocity relative to the air must be added to the wind velocity.

This is the same procedure that was used above for the headwind and the tailwind situations; only now, the resultant is not as easily computed. Since the two vectors to be added - the southward plane velocity and the westward wind velocity - are at right angles to each other, the Pythagorean theorem can be used.

This is illustrated in the diagram below. In this situation of a side wind, the southward vector can be added to the westward vector using the usual methods of vector addition.

The magnitude of the resultant velocity is determined using Pythagorean theorem. The algebraic steps are as follows:. The direction of the resulting velocity can be determined using a trigonometric function. Since the plane velocity and the wind velocity form a right triangle when added together in head-to-tail fashion, the angle between the resultant vector and the southward vector can be determined using the sine, cosine, or tangent functions.

The tangent function can be used; this is shown below:. If the resultant velocity of the plane makes a Like any vector, the resultant's direction is measured as a counterclockwise angle of rotation from due East. The effect of the wind upon the plane is similar to the effect of the river current upon the motorboat. If a motorboat were to head straight across a river that is, if the boat were to point its bow straight towards the other side , it would not reach the shore directly across from its starting point.

The river current influences the Small Chris Craft Boats Pdf motion of the boat and carries it downstream. The resultant velocity of the motorboat can be determined in the same manner as was done for the plane.

The resultant velocity of the boat is the vector sum of the boat velocity and the river velocity. Since the boat heads straight across the river and since the current is always directed straight downstream, the two vectors are at right angles to each other.

Thus, the Pythagorean theorem can be used to determine the resultant velocity. What would be the resultant velocity of the motorboat i. The magnitude of the resultant can be found as follows:. The direction of the resultant is the counterclockwise angle of rotation that the resultant vector makes with due East. This angle can be determined using a trigonometric function as shown below.

Motorboat problems such as these are typically accompanied by three separate questions:. The first of these three questions was answered above; the resultant velocity of the boat can be determined using the Pythagorean theorem magnitude and a trigonometric function direction.

The second and third of these questions can be answered using the average speed equation and a lot of logic.

The solution to the first question has already been shown in the above discussion. We will start in on the second question. The river is meters wide. That is, the distance from shore to shore as measured straight across the river is 80 meters. The time to cross this meter wide river can be determined by rearranging and substituting into the average speed equation.

The distance of 80 m can be substituted into the numerator. But what about the denominator? What value should be used for average speed? With what average speed is the boat traversing the 80 meter wide river? Most students want to use the resultant velocity in the equation since that is the actual velocity of the boat with respect to the shore. And the diagonal distance across the river is not known in this case.

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