• anonymous
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.
  • Stacey Warren - Expert
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  • jamiebookeater
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  • 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!
  • 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
  • matt101
Good point, @IrishBoy123! To be honest I didn't think much of the assumption provided, but you've turned me around.

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