Scooter I said:Homework Statement
A river is flowing 4.0 m/s to the east. A boater on the south shore plans to reach a dock on the north shore 30.0 Degrees downriver by heading directly across the river. What should be the boat's speed relative to the water?Homework Equations
Not sure.
The Attempt at a Solution
I have no idea whatsoever how to do this.
cepheid said:Your velocity relative to the bank is a vector sum of your velocity relative to the water (which is northward, i.e. across the river) and the velocity of the water relative to the bank (which is eastward). Since these two vectors are perpendicular, they form a right triangle, with the resultant (sum) as the hypotenuse. The angle of this diagonal path across the river is going to depend on the relative lengths of the two legs of the right triangle. In other words, it's going to depend on how much the magnitude of the "downstream" velocity compares to the magnitude of the of the "across the river" velocity. The former is given, while the latter is what you're trying to solve for.
So basically you have a right triangle with one of the two perpendicular sides, and one angle. Solving for the other perpendicular side should be trivial.
Scooter I said:How would you solve for it?
Would it be sin30 = 4/V. V = 8?
cepheid said:Well, first of all, I'm assuming that the 30 degrees is measured from due north (i.e. the bearing of your path ends up being 30 degrees east of north).
So the side of the triangle that is opposite the angle is the eastward vector with magnitude 4 m/s.
The side of the triangle that is adjacent to the angle is the northward vector whose magnitude you are trying to solve for.
Which trigonometric ratio is the ratio of these two sides? Hint: it's not sine. Sine involves the opposite side and the hypotenuse, which you're not trying to solve for.
This is basic trig, so if you're feeling shaky about it, I would brush up.
Scooter I said:I think I understand now. Would it be tan30 = (4.0m/s)/(Vb/w) Vb/w = 6.93 m/s?
Also, how would you answer the second part?
cepheid said:Seems ok. For the second part, as I mentioned before, the speed of the boat relative to the shore is the speed of the boat relative to the water + the speed of the water relative to the shore. So for the second part, you want to compute the magnitude of the vector sum of the velocities.
Scooter I said:So Vb/d (where dock is d) = Vb/w + Vw/d
Vb/d = 6.93m/s + 4.0m/s
Vb/d = 10.93 m/s
Correct?
cepheid said:NO, not at all. Did you forget about the fact that we're doing a vector sum and that we therefore have a right triangle? The 6.93 m/s and 4.0 m/s velocity vectors are perpendicular to each other, so their magnitudes don't add directly the way scalars do.
Which side of the triangle represents the magnitude of the vector sum of the northward and eastward velocities? Given the geometry, how would you compute the length of that side?
A further hint: the boat's velocity relative to the bank is the "actual" direction of the boat across the river that would be seen by an observer sitting on the bank. It is neither straight north nor straight east.
Scooter I said:Ah, the magnitude of the vector sum is the hypotenuse given by the formula Vb/d[itex]^{2}[/itex] = Vb/w[itex]^{2}[/itex] + Vw/d[itex]^{2}[/itex].
Vb/d = √(4.0m/s)[itex]^{2}[/itex] + (6.93m/s)[itex]^{2}[/itex]
Vb/d = 8.0 m/s
Correct?
PS: Thanks for all your help, I really really do appreciate it.
cepheid said:Yeah, that looks okay to me. Glad to be of help.
Relative velocity in the context of river flow refers to the speed and direction at which a body of water is moving in relation to an observer or another object. It takes into account both the velocity of the water itself and any additional velocity due to external factors such as wind or the rotation of the Earth.
The relative velocity of river flow is calculated by taking the vector sum of the velocity of the water and the velocity of any external factors, such as wind or the Earth's rotation. This can be represented mathematically using vector addition or graphically using a velocity triangle.
Considering relative velocity in river flow is important because it helps us understand how the water is moving and how it may impact other objects or structures in or near the river. It also allows us to make more accurate predictions about the behavior of the river and potential hazards.
The relative velocity of a river can greatly impact boats or swimmers in the water. If the river is flowing in the same direction as the boat or swimmer, it can increase their speed and make it easier to move downstream. However, if the river is flowing in the opposite direction, it can create strong currents that make it difficult to move or even pull the boat or swimmer in the opposite direction.
Yes, relative velocity in a river can change over time due to various factors such as changes in wind speed or direction, changes in the river's depth or shape, or the addition of new water flows from tributaries. It is important to regularly monitor and calculate relative velocity in a river in order to accurately understand its behavior and potential impacts.