[DiffEq] First order Modeling Applications

[metiscus]

YATP - Yet Another Trainlike Problem

Sailboats A and B each have mass 60kg and cross the starting line at the same time of a race. Each has an initial velocity of 2m/s.
Obviously from this, m1 = m2 == 60kg and vo1 = v02 == 2 m/s.
The wind applies a constant force of 650N to each boat and the water resistance is proportional to the velocity of the boat.
Boat A:
proportionality constants are b1 = 80 Nsec/meter before planing when the velocity is less than 5m/s and b2=60Nsec/m when velocity is aboce 5m/s.

Boat b:
proportionality constants are b1 = 100 Nsec/meter before planing when the velocity is less than 6m/s and b2=50Nsec/m when velocity is aboce 6m/s.

The race is 500m long, which sailboat will be leading at the end of the race?


I assume that the method of solution will be to complete the equations of motion for both then sub them for the race length but the problem that is getting me is the different constants for different times. Do I need the heavyside function or something to get this set-up.

 

[Nate810]

The solution to this question involves solving the equations of motion for both boats using the given information. The equation of motion for each boat is F = ma + bv, where F is the net force, m is the mass, a is the acceleration, b is the proportionality constant, and v is the velocity. Since the wind applies a constant force of 650N, we can substitute F = 650N in each equation. For Boat A, the equation of motion is 650N = 60kg(a) + 80Nsec/m(v) when v < 5m/s and 650N = 60kg(a) + 60Nsec/m(v) when v > 5m/s. For Boat B, the equation of motion is 650N = 60kg(a) + 100Nsec/m(v) when v < 6m/s and 650N = 60kg(a) + 50Nsec/m(v) when v > 6m/s.Once these equations are solved, they can be integrated with the initial conditions (initial velocity = 2m/s) to obtain the position of both boats over the 500m race. Whichever boat has a higher final position at the end of the race will be leading.

 

[quicknote]

I would approach this problem by first identifying and defining all the relevant variables and equations. In this scenario, the variables are the mass (m), initial velocity (vo), force (F), and velocity (v) for each boat. The equations that govern their motions are Newton's Second Law (F=ma) and the equation for velocity (v=vo+at).

Next, I would use these equations to create a system of differential equations, taking into account the different proportionality constants (b1 and b2) for each boat at different velocities. This would result in a set of first-order differential equations that can be solved using numerical methods or through analytical solutions.

To determine which sailboat will be leading at the end of the race, I would use the equations to simulate the motion of both boats over the 500m race distance. This would allow for a comparison of their velocities at the finish line, and the boat with the higher velocity would be the winner.

In terms of the different constants for different times, it is important to consider the physical properties of the sailboats and how they affect the resistance from the wind and water. The heavyside function may be useful in setting up the equations, as it can be used to switch between different constants depending on the velocity of the boat.

In conclusion, this problem can be solved by using the principles of differential equations and numerical methods to simulate the motion of the sailboats and determine which one will be leading at the end of the race. It is important to carefully consider all the variables and equations involved and to use appropriate techniques to account for the different constants at different velocities.