Find Minimum Time for Crossing River Using Powerboat: 2D Motion Problem

[joe215]

Homework Statement

A hunter wishes to cross a river that is 1.5 km wide and flows with a speed of 5.0km/h parallel to its banks. the hunter uses a small powerboat that moves at a maximum speed of 12 km/h with respect to the water. What is the minimum time necessary for crossing?

The Attempt at a Solution

5km/h^2 + 12Km/h^2=sq rt 169km/h^2 = 13 km/h

The hunter moves really moves at 13 km/h to the right because of the water moving upstream. 13 is the vector resultant of 5 north and 12 east.

So, T=D/V

T=width of stream/velocity of hunter
T=1.5km/12km/h = 0.125 h

 

[anathema]

= 7.5 minutes

Thank you for your interesting question. I would approach this problem by first understanding the principles of vectors and relative velocities. The hunter's boat will have a resultant velocity of 13 km/h (as you correctly calculated) due to the combination of the water's velocity and the boat's maximum speed.

To find the minimum time necessary for crossing, we can use the formula T = D/V, where T is time, D is distance, and V is velocity. In this case, we are looking for the time it takes to cross a distance of 1.5 km with a velocity of 13 km/h.

Therefore, T = 1.5 km / 13 km/h = 0.115 hours or 6.9 minutes. This is slightly different from your calculation of 7.5 minutes, but it is important to note that we should use the resultant velocity of 13 km/h instead of just the boat's maximum speed of 12 km/h.

I hope this helps to clarify the solution. Let me know if you have any further questions.

 

[baba89]

or 7.5 minutes

I would like to clarify that the hunter's motion in this problem is not a 2D motion, but rather a 1D motion along the river. The motion of the powerboat is not affected by the river's flow since it is traveling parallel to the banks. Therefore, the minimum time for crossing the river can be found by dividing the width of the river by the relative velocity of the powerboat, which is 12 km/h. This gives a minimum time of 0.125 hours or 7.5 minutes, as calculated correctly in the attempt at a solution. However, it is important to note that this is the minimum time for crossing if the powerboat is able to maintain its maximum speed of 12 km/h throughout the entire journey. Any decrease in speed due to external factors, such as wind or currents, would result in a longer crossing time.