I've been working on this one and I just can't figure it out. I appreciate all the help:

Two miles upstream from his starting point, a canoeist passed a log floating in the river's current. After paddling upstream for one more hour, he paddled back and reached his starting point just as the log arrived. Find the speed of the current.

I think it may be one of those two-equation problems you need to set up in order to get the answer but I don't know where to start. Thanks for helping me.

3 answers

1/2 mph = current

1 mph

To solve this problem, we can use the concept of relative velocity. Relative velocity is the combined effect of the velocities of two objects as seen from one of the objects. In this case, we need to consider the velocity of the canoeist relative to the log.

Let's assume the speed of the canoeist in still water is C (in miles per hour) and the speed of the current is V (in miles per hour).

When the canoeist is paddling upstream, his effective speed with the current is reduced. So, relative to the log, the speed of the canoeist will be C - V, since the current is aiding the log in moving downstream.

When the canoeist is paddling downstream, his effective speed is increased. So, relative to the log, the speed of the canoeist will be C + V.

Now, let's apply this information to the problem:

The canoeist passes the log when he is two miles upstream from his starting point. This means that he can cover the distance between the log and his starting point in one hour paddling downstream, and in two hours paddling upstream.

Using the relative velocities, we can write the following equations:

Distance = Speed × Time

For the downstream paddling:
2 miles = (C + V) × 1 hour ----- Equation 1

For the upstream paddling:
2 miles = (C - V) × 2 hours ----- Equation 2

We now have two equations with two unknowns (C and V). We can solve these equations simultaneously to find the speed of the current.

Let's proceed with solving the equations:

From Equation 1, we have:

2 = C + V

From Equation 2, we have:

2 = 2(C - V)
2 = 2C - 2V
2C - 2V = 2
Divide the entire equation by 2 to simplify it:
C - V = 1

Now we have a system of linear equations:

C + V = 2
C - V = 1

We can solve this system of equations by adding the two equations together:

(C + V) + (C - V) = 2 + 1
2C = 3
Divide both sides of the equation by 2:
C = 3/2
C = 1.5 mph

Now that we've found the speed of the canoeist in still water, we can use Equation 1 to calculate the speed of the current:

2 = (1.5 + V) × 1
2 = 1.5 + V
2 - 1.5 = V
0.5 = V

Therefore, the speed of the current is 0.5 miles per hour.

The canoeist's speed in still water is 1.5 mph, and the speed of the current is 0.5 mph.