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Upstream Downstream Boat Math Problem Dd = distance gone downstream Du = distance gone upstream. As with other uniform motion word problems in Algebra, an important algebraic equation to remember is: Distance = Rate * Time. With the boat-in-the-river math problem, we modify this slightly when we are going downstream to get: Dd = (B + W) * Td. That is, the speed (or rate) of the boat going downstream is the speed of the boat in still water plus the speed of the water (or river .
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When a boat is going upstream, the river works against you so we have to change our distance equation to reflect this by subtracting the speed of the water from the speed of the boat to get our rate:. As always with learning about how to solve these kinds of problems, it is always best to see step by step how to solve specific examples.

Sample Boat-in-the-River Word Problem 1: A boy can row downstream 18 miles in 2 hours, but it will take him 6 hours to return. How fast can he row in still water, and what is the rate of the current?

I always recommend starting by writing down the two essential algebraic distance equations to start:. The water is flowing at a rate of 3 miles per hour, while the boat is moving in still water at a rate of 6 miles per hour. Sample Boat-in-the-River Word Problem 2: A crew that can row 12 mph downstream finds that 2 miles downstream takes the same time as 1 mile upstream.

Find the rate of the current. Man starts Boat Travel Upstream And Downstream Problems In from A, travels to B and comes back. Therefore, Upstream Downstream Boat Problems Quote above equation becomes,. Also we have calculated downstream and upstream speeds at the start see values 1 and 2. In question, you can see that the man takes 40 minutes to travel to B and come back to A. You have to convert this to hours and apply in above equation. We are converting from minutes to hours because we are using speed values in km per hour units. It takes him twice as long to row up as to row down the river.

Find the rate of the stream. Solution: Step 1: Calculate upstream and downstream speeds. Based on our assumptions, you can easily calculate upstream and downstream speeds as shown below. In this type, you have to form linear equations based on conditions given. You have to solve those equations to find the answer. Example Question 5: Kavin can row 10 km upstream and 20 km downstream in 6 hours.

Also, he can row 20 km upstream and 15 km downstream in 9 hours. Find the rate of the current and the speed of the man in still water.

Solution: You have to make below assumptions to form equations. You already know the below equation. 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 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. 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.

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