On occasion objects move within a medium that is moving with respect to an observer. 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 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 Boat And Stream Formula In Bengali Outlook as these are typically accompanied by three separate questions:.
The first of these three questions was answered above; Boat And Stream Formula In Bengali Web 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. If one knew the distance C in the diagram below, then the average speed C could be used to calculate the time to reach the opposite shore.
Similarly, if one knew the distance B in the diagram below, then the average speed B could be used to calculate the time to reach the opposite shore. And finally, if one knew the distance A in the diagram below, then the average speed A could be used to calculate the time to reach the opposite shore. It requires 20 s for the boat to travel across the river. During this 20 s of crossing the river, the boat also drifts downstream.
Part c of the problem asks "What distance downstream does the boat reach the opposite shore? And once more, the question arises, which one of the three average speed values must be used in the equation to calculate the distance downstream? The distance downstream corresponds to Distance B on the above diagram.
The speed at which the boat covers this distance corresponds to Average Speed B on the diagram above i. The mathematics of the above problem is no more difficult than dividing or multiplying two numerical quantities by each other. The mathematics is easy! The difficulty of the problem is conceptual in nature; the difficulty lies in deciding which numbers to use in the equations.
That decision emerges from one's conceptual understanding or unfortunately, one's misunderstanding of the complex motion that is occurring. The motion of the riverboat can be divided into two simultaneous parts - a motion in the direction straight across the river and a motion in the downstream direction.
These two parts or components of the motion occur simultaneously for the same time duration which was 20 seconds in the above problem. The decision as to which velocity value or distance value to use in the equation must be consistent with the diagram above. The boat's motor is what carries the boat across the river the Distance A ; and so any calculation involving the Distance A must involve the speed value labeled as Speed A the boat speed relative to the water.
Similarly, it is the current of the river that carries the boat downstream for the Distance B ; and so any calculation involving the Distance B must involve the speed value labeled as Speed B the river speed. Together, these two parts or components add up to give the resulting motion of the boat.
That is, the across-the-river component of displacement adds to the downstream displacement to equal the resulting displacement. And likewise, the boat velocity across the river adds to the river velocity down the river to equal the resulting velocity. Now to illustra te an important point, let's try a second example problem that is similar to the first example problem. Make an attempt to answer the three questions and then click the button to c heck your answer.
The resultant velocity can be found using the Pythagorean theorem. It is. An import ant concept emerges from the analysis of the two example problems above.
In fact, the current velocity itself has no effect upon the time required for a boat to cross the river. The river moves downstream parallel to the banks of the river. As such, there is no way that the current is capable of assisting a boat in crossing a river. While the increased current may affect the resultant velocity - making the boat travel with a greater speed with respect to an observer on the ground - it does not increase the speed in the direction across the river.
The component of the resultant velocity that is increased is the component that is in a direction pointing down the river. It is often said that "perpendicular components of motion are independent of each other.
The time to cross the river is dependent upon the velocity at which the boat crosses the river. It is only the component of motion directed across the river i. The component of motion perpendicular to this direction - the current velocity - only affects the distance that the boat travels down the river. This concept of perpendicular components of motion will be investigated in more detail in the next part of Lesson 1. Determine the resultant velocity of the plane magnitude only if it encounters a.
If the width of the river is 80 meters wide, then how much time does it take the boat to travel shore to shore? NOTE: the direction of the resultant velocity like any vector is expressed as the counterclockwise angle of rotation from due East.
If the width of the river is meters wide, then how much time does it take the boat to travel shore to shore? NOTE: the direction of the resultant velocity like any vector is expressed as the counterclockwise direction of rotation from due East.
It would require the same amount of time as before 20 s. Changing the current velocity does not affect the time required to cross the river since perpendicular components of motion are independent of each other. Note that an alteration in the current velocity would only affect the distance traveled downstream and the resultant velocity. Physics Tutorial.
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