A motorboat and its load weigh 2150N. Assuming the propeller force is constant and equal to 110 newtons and water resistance is equal numerically to 6.7V Newton where V is the velocity at any instant in m/s, and the boat starts from rest determine the speed and the distance traveled at the end of 10 seconds.
Answer is 22.72 m/s
If you layout the freebody diagram we should get the equation:
( am i wrong? maybe freebody diagrams are reserved for things that aren't moving)
110 = 6.7(dx/dt)
derive if we get 0 = 6.7*acceleration; which is a dead end.
Integrate then:
110t = 6.7x
plug in 10 seconds in t; I don't get the right answer… What am I supposed to do? Need hint especially how to construct the model/differential equation for this.
Best Answer
Start with Newton's Law of Motion:
$$\sum F=ma$$
The mass of the boat and it's load is $m=\frac{2150}{g}\approx 216.16$
The sum of the forces is $110 - 6.7V$.
Thus, the equation for motion becomes:
$$216.16\frac{d^2x}{dt^2}+6.7\frac{dx}{dt}-110=0.$$
Solve using your favorite technique for constant coefficient nonhomogeneous ODE. (The method of undetermined coefficients should suffice.)
Note that the solution to this equation does indeed have $x(10)=22.722\ m$.
edit:
The velocity equation is first order, linear.
$$216.16\frac{dV}{dt}+6.7V=110$$