Name:_________________ Physics 11 Honour Date:______________

Unit 5 Vectors

5.5 - Applications of Vectors the Across the River Problem 2-D Relative Motion

Moving a boat across a river is basically a "relative velocity" problem in two directions. We start with the

following assumptions to orient ourselves (note : essentially, we are picking our coordinate system - we can

pick it any way we want, we just need to be consistent for the rest of the problem) :

One other thing that we will need to keep in mind as we begin to solve these problems is that the

components are mutually independent.

This might sound like an oxymoron, since mutually means together, but independent means apart.

What it means is that the two components are obviously working together (mutually),

touching head-to-tail and stuff, but they are still measuring separate things (independent).

If one of the components changes it will not affect the other. For example, if x got bigger, y would still

be the same. Only the resultant would change.The Across the River Problem

Let's say you point your boat directly East across a river that is flowing North.

As your boat moves itself through the water towards the East, the river is constantly pushing it

to the North.

In the end someone watching from the shore would say that you were moving in a direction roughly

North-East.

This happens because the two components mutually work together to give us our resultant.

One way to label the vectors with subscripts is shown here

bvw = velocity of the boat in still water ( Velocity of ____________ relative to ____________ )

wvs = velocity of the water with respect to the shore

bvs = velocity of the boat with respect to the shore

Example 1

You are rowing a boat across a river, aiming directly for the other side. You are rowing at 3 km per hour but the river is flowing at 4 km per hour. If the river is 200 m wide,

a) how long will it take the boat to cross the river? b) how far down-river will you land on the other side?

Example 2

An airplane is pointing straight east and flying with airspeed of 300 km/hr. there is a southerly wind of 80 km per hour. What is the actual direction of the flight of the plane and what is its ground speed?

Example 3

You are pedaling your bike at 20 km per hour along a road running to the east. There is a northerly wind of 15 km per hour. From which direction does the wind appear to be coming toward your face?

Example 4

An airplane is heading in the direction 20 degrees North of East. Its airspeed is 400 km per hour. There is a 120 km per hour wind from the direction 30 degrees south of East. What is the ground-

speed and the direction of the flight of the plane?

Example 5: A swimmer heads directly across a river swimming at 1.6 m/s relative to still water. She arrives

at a point 40 m downstream from the point directly across the river, which is 80 m wide. Determine …

a) the speed of the current,

b) the magnitude of the swimmer's resultant Velocity

c) The direction of the swimmer's resultant velocity

d) The time it takes the swimmer to cross the river

5.5 Assignments 1. A boat can travel 2.30 m/s in still water. If the boat heads directly across a river with a current of

1.50 m/s:

a. What is the velocity of the boat relative to the shore? 2.75 m/s

b. At what angle compared to straight across is it traveling? = 33.1˚

c. How far from its point of origin is the boat after 8.0 s? 22 m

d. At what upstream angle (compared to straight across) must the boat travel in order to the other bank

directly opposite its starting point? How fast across the stream is it traveling? = 40.7˚, 1.74 m/s

2. You are flying a hang glider at 14 km/h in the northeast direction (45°). The wind is blowing at 4 km/h from due north.

a) What is your airspeed?

b) What angle (direction) are you flying?

c) The wind increases to 14 km/h from the north. Now what is your airspeed and what direction are you flying? If your destination is to the northeast, how would you change your

speed or direction so you might make it there? Test your answer using the sim.

Vector Problems (Trig. Solutions)

1. How far East has a person walked if he travels 350 m in a direction 25 E of N? 148 m

2. What would be the resulting displacement if a snail crawls 2.0 m north and then 3.0 m east? What is the

snail's direction from the starting point? 3.6 m & 33.7˚ N of E

3. Find the magnitude and direction from the horizontal of a 40.0 N upward force and 17.0 N horizontal

force. 43.5 N at 67˚

4. A boat travels east at 13 km/hr when a tide is flowing north at 1.2 m/s. Find the actual velocity and

heading of the boat. 3.8 m/s at 18.4˚ N of E

5. A person that swims at 3.2 m/s swims straight across a river with a current of 1.4 m/s. What is the

resulting velocity of the swimmer (across and down stream)? At what angle compared to straight across

is the swimmer moving? 3.5 m/s at 23.6˚

6. The swimmer above decides to swim into the current at such an angle that he will travel straight across.

Find the angle (compared to straight across) at which he would have to swim. Calculate the velocity

across the stream. 2.9 m/s at 25.9˚

Vector problems (Component or Sine-Cosine Law Solutions)

1. seagull flying with an air speed of 10.0 km/h is flying north but suddenly encounters a wind of 5.0 km/h

at 20 south of east. What will be the new direction and airspeed of the seagull? 9.5 km/h at 60.5˚ N of E

2. A pilot wishes to reach a city 600.0 km away in a direction of 15 S of W in two hours. (v = 300 km/h at

this same direction - this is the resultant vector in the vector diagram!)

If there is a wind of 70 km/h blowing at 10 W of S. What must be the heading and air speed of the

plane? heading is 2˚ S of W at an airspeed of 278 km/h

3. A plane that can fly at 250 km/h wishes to reach an airport that has a bearing of 25 W of N from its

present location.

a)If there is a 50.0 km/h wind blowing directly to the west what should be the heading of the plane. 14.6˚ W of N (set up vector diagram and then use Sine Law)

c) What will be its ground speed? 267 km/h

c) How long would it take to get to the airport if it were 560 km away? 2.1 hours