physics problem

Question

2) A boat traveling west encounters a current moving 8m/s south. The boat travels 56m downstream while crossing the 100m wide river.
a) How much time does it take to travel shore to shore?
b) Calculate the boat's velocity relative to the water.
c) What is the boat's resultant velocity relative to an onshore observer?

Answer 1

Hi Jessica.

 

a) If the current is moving 8 m/s South, that means the boat is also moving in the southbound direction at that speed.

 

The boat traveled 56m south in the time it took to cross the river. So:

 

Distance = Rate*Time

 

56 m = 8 m/s * t

 

t = 7 seconds.

 

b) The boat's velocity relative to the water is simply how fast it's moving shore to shore. You know from the problem that the river is 100m wide, and you know from a) that it took 7 seconds to cross.

 

Distance = Rate*Time

 

100 m = Rate*7 s

 

Rate = 100/7 m/s, ~ 14.28 m/s

 

c) The boat's velocity relative to an onshore observer:

 

This is a bit different from b), because if you're floating in the water watching the boat move, all it appears to be doing is moving forward, so all you need to figure out is the shore-to-shore velocity.

 

But, if you're stationary on shore, the boat appears to be moving both shore-to-shore and downstream. This is where you need to break out your vectors to find out how fast it appears to be moving.

 

Shore-to-shore velocity: 100/7 m/s

Downstream velocity: 8 m/s

 

Let's call the resultant velocity V_r

 

Since the velocities are perpendicular, the vector addition is simple. Just imagine the magnitudes as the lengths on a right triangle.

 

(V_r)^2 = (100/7)^2 + (8)^2

 

V_r = ~16.37 m/s

 

Ugly numbers, but, there it is.