## anonymous one year ago Hi everyone. Anybody have an idea to point me in the right direction? A motorboat traveling at a speed of 2.2 m/s shuts off its engines at t=0. How far does it travel before coming to rest if it is noted that after 2.9 s its speed has dropped to half its original value? Assume that the drag force of the water is proportional to v.

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1. matt101

The question gives us the following information: Initial velocity = vi = 2.2 m/s Final velocity = vf = 0.5vi = 1.1 m/s Time = t = 2.9 s We can also assume constant acceleration since the drag force is proportional to v. Using this information, you should be able to use equations of motion to find the distance travelled!

2. IrishBoy123

because drag $$\propto v$$, we can say $$F = ma = - kv$$ where k is some constant that creates the differential equation $${dv \over dt} = - {k \over m} v$$ which we solve as $$\large \int\limits_{v=v_o}^{v(t)} {1 \over v} dv = -\alpha \int\limits_{t=0}^{t} dt$$ where $$\alpha = {k \over m}$$ [because we know neither the value of m or k, we may as well stuff them into a single constant called $$\alpha$$] your solution will be in the form of exponential decay, and then to solve for $$\alpha$$, just plug in the various values we have been given for v and t. *however* please note that the boat in theory never actually stops. exponential decay is asymptotic....maybe you are supposed to spot this at the outset, i dunno, seems strange to me but this is the correct analysis :p

3. matt101

Good point, @IrishBoy123! To be honest I didn't think much of the assumption provided, but you've turned me around.