Max Boat Speed w/ 2 Engines: Proportional to v^2 Resistive Force

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The discussion centers on calculating the maximum speed of a boat with two engines, given that the resistive force is proportional to the square of the velocity (v^2). When one engine is at full throttle, the boat reaches a maximum speed V, and using two engines theoretically doubles the force available. This leads to the conclusion that the maximum speed with two engines would be greater than V, but the exact relationship requires understanding the balance of forces. The resistive force equation complicates finding the maximum speed, as it involves nonlinear dynamics, but basic principles suggest that the speed increases with additional engine power. Ultimately, the key takeaway is that while advanced calculus may be needed for detailed motion equations, the maximum speed can be reasoned through basic force balance concepts.
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Here is a question that I came across while doing my problem set questions:

A boat experiences a resistive force that is proportional to v^2. If the maximum speed of a boat is V when one engine is at full throttle, what would be the maximum speed if two engines were used?

I assume this so-called resistive force is like air resistance, but when I searched up the formulas, they're absolutely long and I will never be able to solve those with my current math skills.

The way I tried to see this problem is that if it is related to energy or something, can anyone help me here?
 
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You need to balance forces. Finding the equation of motion is very difficult because the v^2 term makes the equation of motion a non linear ODE.
 
The maximum speed implies forces are balanced , with the boat moving at constant velocity, that is, the force delivered by the engine equals the resistive force, kv2. Two engines can deliver twice the force. What does this imply about the maximum speed when 2 engines are used? You would need a good knowledge of calculus to find the equation of motion for the speed at any time t, but you don't need any advanced math to solve for the maximum speed.
 
Edit: nvm, didn't read the post above.
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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