A boat pushes off from a pontoon and travels 5m/s East while the pontoon floats away at 2m/s south. Calculate the velocity of the boat relative to the pontoon????? .

If you don't understand the vector concept, go to Method B.

Method A: vector method.

Let
#vecV_b = 5m/s hatx#, velocity of the boat
#vecV_p = -2m/s haty# velocity of the pontoon

#vecV_(b p) = #Velocity of the boat relative to the pontoon#

Relative velocity means comparing #vecV_b# to #vecV_p#, i.e., the difference of #vecV_b# from #vecV_p#

#vecV_(b p) =vecV_b - vecV_p = 5m/s hatx -(-2m/s haty)=5m/s hatx +2m/s haty #,

The magnitude of #vecV_(b p)# is:

#V_(bp) = sqrt((5m/s)^2+(2m/s)^2)=5.4 m/s#

The direction of #vecV_(b p)# is:

#theta = tan^-1(2/5) = 22^@ #

Method a) Reasoning without using vector concept

Method B: reasoning without using vectors.

Let
x = distance traveled by the boat t seconds due E after the pushing
y = distance traveled by pontoon t seconds due S after the pushing

At t=1.0 s later,

x= 5 m East
y = 2m South.

x and y form an right angle triangle.

Then distance (d) between them (from center of mass of the boat to the center of mass to the pontoon if you want) is

# d = sqrt(x^2 + y^2) =sqrt(5^2+2^2) = 5.4 m#

The rate of this separation is therefore

#d/t = (5.4m)/(1s) = 5.4 m/s#

Hence the boat is moving away relative to the pontoon at a speed 5.4 m/s.

To find which direction the boat is moving away relative to the pontoon, you have to imagine that you are sitting on the pontoon facing south. To see which direction the boat is moving away, you have to turn your head to the east, then northward. Hence the boat is traveling in the NE direction.

The angle, illustrated in the figure above, is thus:

#theta = tan^-1 (y/x) = tan^-1(2/5) = 22^@ # North of East (NE)

(Note, the answer can be simply written as #22^@# because that is how a positive angle is defined: measured couter-clockwisly from the East or the x-axis, which is zero degree. Counterclockwise angle is marked as positive.

Finally,
the velocity of the boat relative to the pontoon:
It has a speed 5.4 m/s and at an angle #22^@#,