A man can swim at 4 ft/s in still water. He wishes to cross tje 40-ft wide river to point B, 30 ft downstream. If the river flows with a velocity of 2 ft/s, determine the speed of the man and the time needed to make the crossing. Note While in the water he must not direct himself toward point B to reach the point.

How do you solve for answers:
Vm = 4.87 ft/sec
t = 10.3 sec

This is a vector problem. Draw a picture of the river going up and down. Now put a dot on the left-hand side of the river to represent the man. Draw a dot on the right-hand side lower on the page to represent where he wants to go. Draw a horizontal line going from the top left dot straight across the stream -- this is 40ft. Label the distance from there down to the bottom-right dot 30 feet. Now draw a line connecting the upper right dot to the lower right dot and label that 50 feet (from the pythagorous theorem). We can also figure out the angle between the horizontal line you drew and the 50 feet line. This is 36.86989765... degrees.

Now I'm going to have you draw another picture separate from the river picture. Draw a horizontal line; label the left point A and the right point B. Now go straight down from B for a little bit and put another point, C. Then, keep going down and get a 4th point, D. Draw a line from A to C and a line from A to D.

The vector from A to C represents the speed and direction he should swim. He's not going to swim straight across or straight down, but we don't know what direction he should actually swim. We don't know what the angle from AB to AC is, so it's just a guess. AD represents the speed and direction he'll actually be moving. BD, then, represents the speed he'll be moving downstream. AB represents the speed he'll be moving horizontally across the stream.

Label AC 4, because he can swim at 4 feet/sec. CD is then 2 -- he travels at 2ft/s downstream from the current of the river. These two vectors add together to make AD, the direction he moves. Does this make sense? You can also add that the angle from AB to AD is 36.87 degrees.

We have some unknowns here. The speed with which he actually travels, AD, is what we are trying to find. Let's call that x. The other speed we don't know is BC, how fast his swimming contributes to him moving downstream. Let's call that y. We also don't know AB, the speed he moves horizontally. Let's call that z.

We can now write several equations. First, from the pythagorous theorem, z2 + y2 = 42 and z2 + (y+2)2 = x2. We also know that sin(36.87) = 30/50 = (y+2)/x. You now have a system of three equations with three variables. Solving those gives you x = 4.87 ft/sec

The final step is figuring out how long it takes him. You know he is moving at a speed of 4.87 ft/sec, and you know the distance is 50 feet. 50 feet * 1sec / 4.87 feet = 10.3 sec

energy can cross the boundaries of a closed system in the form of heat and mass

explanation: