Question

A toy submarine of mass m = 0.88 kg moves around a submerged circular track of radius R = 1.72 m. The submarine\'s engine provides a constant propulsion force of F = 5.08 N. When the sub is in motion, it is subject to a viscous drag force exerted by the water. This force is proportional to the sub\'s speed; the proportionality factor is C = 1.01 kg/s. Assuming it starts from rest at t = 0 s, the speed of the submarine at a later time t is given by

Answer 1

\[V（t)=\frac{F}{C}(1-e ^{-\frac{ Ct }{ m }})\]

Answer 2

where e is the base of the natural logarithm

Answer 3

How much time has passed when the submarine\'s speed reaches 63% of its terminal value?

Answer 4

@ganeshie8

Answer 5

@theEric , help me plz!!!

Answer 6

hi @theEric , I want o find the ime taken for the 63% of terminal value to be reached. SO I find the value of time taken for the 100% terminal value to be reached. SO I take the time derivative of V（t) and set that =0, since if the submarine is at terminal velocity, no acceleration. so dv/dt must be 0. However, the equation does not have a 0 solution

Answer 7

I'm sorry! I've forgotten too much of ordinary differential equations to be helpful!

Answer 8

I couldn't get to that first part, to be honest.

Answer 9

I think you're approach is correct, but I can't be sure. I've done problems like these, but I've forgotten since I have not practiced them.

Answer 10

I know the terminal velocity will be where dv/dt is 0, though. But my understanding is nearly insignificant for this problem at this time.

Answer 11

I'm sorry I can't help!

Answer 12

It's helpful to have a picture of what the speed looks like as a function of time, roughly

How to see the terminal value of v ? Look at the exponential term in the expression for v and notice that as t gets larger, this term gets smaller, until for t 'very large', whatever that means, the exponential term is practically zero. Clearly, v is approaching a value of F/C as t goes to infinity.

Now you just need to know the value of t for which v = 0.63F/C.

You can just go ahead and solve the equation, taking logs etc, or you can notice that 1/e is approximately 0.37, which means that the particular value of t that you require occurs when Ct/m = 1.

This kind of curve comes up from time to time in physics problems, so it's worth being familiar with it.