course_planning:183_projects:f21_week_1_problem_river_crossing_problem_solution

• Perform the following mathematical operations on (physical) vector quantities: vector addition/subtraction, magnitude/unit vector.
• Sketch vector quantities and perform graphical (physical) vector addition/subtraction.
• Predict the motion of a single-particle system executing constant velocity or constant acceleration motion using appropriate representations (this includes verbal, graphical, diagrammatic, mathematical, and computational representations).
• Breaking vectors into their components
• Relative velocity
• Position update formula - iterative motion
• Familiarizing yourself with VPython
• What is a vector and what quantities in the readings this week were vectors?
• What is a unit vector?
• What is the difference between distance and displacement?
• What is the difference between speed and velocity?
• What is the relative velocity equation?

# Project 1: Part A: Riverboat Crossing

You are a group of scientists and engineers who are studying the Miskatonic River. You need to get across the river to take more measurements but your boat sprung a leak and sank. At the edge of the river which you have found to have a constant flow rate you come across a child and their parent playing with a remote control boat. There is also an old motorboat that you could use but its rudder is stuck in place and the motorboat is positioned so that it is facing directly perpendicular to the river flow. The old motorboat indicates that it has a top speed of 7m/s. On the other side of the river, there are a lot of sharp rocks and the river is very strong so you need to know where you are going to make landfall on the other side before you start the boat’s engine. In your possession, you have a laser distance measure and a stopwatch. You use the laser distance measure to find that the river is 20 meters wide but there must have been very little battery power left as the measure stops working after measuring the width of the river. You also notice that on your side of the river there are two docks. One that has the motorboat and the other that is down the river that has nothing docked on it. If you turn on the boat when it is positioned directly perpendicular to the river flow calculate where you will end up on the other side of the river.

# Project 1 Solution: Part A: Riverboat Crossing

The students need to identify that the crux of the problem is finding the speed of the river. To do so, the students need to identify that they should do 3 experiments with the RC boat.

Here are the links to the experiments:

Across River

Dock to Dock Forward

Dock to Dock Backwards

The first is sending the RC boat across the river. By doing this you are able to find the velocity of the RC boat with respect to the water:

$$\frac{River Width}{Tacross} = V_{RC/R}$$

$V_{RC/R}$ should be 4.7 m/s but with the times I recorded I typically got lower than this)

They should then set up two equations with three unknowns. The variables are as follows:

$V_{RC/R}$ = RC Boat Velocity with respect to river - this is known as you just calculated it.

$V_{R/S}$ = River Velocity respect to shore - unknown

X = Distance between docks - unknown

T1 = time with the river - known (once they time it) - sending the boat from one dock to the other with the river and timing amount of time it takes.

T2 = time against river - known (once they time it) - sending the boat from one dock to the other against the river and timing the amount of time it takes.

$$V_{RC/R} + V_{R/S} = \frac{X}{T1}$$ (Important for students to realize that $\frac{X}{T1}$ is $V_{RC/S}$ going with the river)

$$V_{RC/R} - V_{R/S} = \frac{X}{T2}$$ (Important for students to realize that $\frac{X}{T2}$ is $V_{RC/S}$ going against the river)

At this point, they should recognize that they need to do two additional experiments. Send the boat between the two docks with the river and record the time it takes to travel the distance. Then send the boat between the two docks against the river and record the time it takes to travel the same distance.

You find values for: T1 and T2

Solve for the river velocity by getting an unknown variable by itself - for example:

$$X = T1*V_{RC/R} + T1*V_{R/S}$$

Substitute into the second equation

$$V_{RC/R} - V_{R/S} = \frac{(T1*V_{RC/R} + T1*V_{R/S} )}{T2}$$

$V_{R/S}$ should be the only unknown at this point. It should be 2m/s but the students’ answers will be highly dependent on their time measurements which can create large discrepancies.

X should be 20 but when I solved for it using the times I recorded I got 29 so again there can be a big + or - on the value.

Knowing the river velocity and knowing that the boat is aimed straight across we then need to solve for the distance boat travels down the river. To do so we need to divide the length of the river, 20 meters by 7m/s to find how long it takes the boat to cross the river. Then multiply this number by $V_{R/S}$. (This should be somewhere around 10 but again highly variable depending on their measurements).

##### Tutor Questions
• Question: How did you find the speed of the boat with respect to the water?
• Expected Answer: …we sent the boat perpendicular to the shore…
• Question: How did you know to send the boat perpendicularly across the river?
• Expected Answer: …the only distance we have to use in $x=v\,t$ is the perpendicular distance $W$…
• Question: Why is $x=v\,t$ applicable here?
• Expected Answer: …the velocity is constant…
• Question: How did you find the velocity of the river?
• Expected Answer: …we sent the boat up and down the shore…
• Question: How did you know to send it up and down the river?
• Expected Answer: …we can use relative velocity $\vec{v}_{\rm a/c}=\vec{v}_{\rm a/b}+\vec{v}_{\rm b/c}$…