Solving Differential Equation: Boy & Girl's Meeting Time

In summary, the conversation discusses a problem involving a boy on a boat and a girl on the shore. The boy moves towards the girl with a constant velocity while keeping the boat pointed towards her. The equations and solution method for finding the time of their meeting are provided, and a figure is discussed to better understand the problem. The discussion also includes a suggestion for a standard substitution method to solve the integral.
  • #1
GregoryGr
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Homework Statement



A boy is on a boat, at a distance H from the shore, when he sees a girl (at the point on the shore where the distance is measured) running with a constant velocity u parallel to the shore. At that time, he moves towards her, with a speed v, in such a way, that the point of the boat is always pointed at the girl (so his vector is always pointing her way). Find the time of their meeting.


Homework Equations



$$ \vec{v}= \frac{dr}{dt}\hat{r}+r\frac{d\theta}{dt}\hat{\theta} $$

The Attempt at a Solution



I changed the system of coordinates so the girl is stationary at O' , and using polar coordinates since the vector of the boat is always in the radial direction. The velocity for the boat in polar coordinates can be written:
$$ \vec{v}= -(v+ucos\theta)\hat{r}+vsin\theta\hat{\theta} $$

And since the 2 quantities in front of the singular vectors must be the same, I get 2 differential equations which give me this:

$$ \frac{dr}{r}= \frac{-(v-ucos\theta)d\theta}{usin\theta} $$


I am a noob at solving differentials, so firstly, are my equations right, and second, can somebody help me figure out how to do the math? :confused:
 
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  • #2
GregoryGr said:
are my equations right, and second, can somebody help me figure out how to do the math?
Yes, I agree with your first two equations, and I think the third is right. Once you have a solution you can check it satisfies the second one.
If you expand the numerator, the second term is easy to integrate. The first becomes the integral of a cosecant, which is standard enough that you can look it up. If you're interested in doing it from first principles, multiply top and bottom by sine and convert the sin2 in the denominator to 1-cos2. You can then factorise that and expand it using partial fractions. Each fraction is readily integrated.
 
  • #3
GregoryGr said:
I changed the system of coordinates so the girl is stationary at O' , and using polar coordinates since the vector of the boat is always in the radial direction. The velocity for the boat in polar coordinates can be written:
$$ \vec{v}= -(v+ucos\theta)\hat{r}+vsin\theta\hat{\theta} $$

I think there should be "u" instead of v in front of sinθ. Check.


ehild
 
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  • #4
ehild said:
I think there should be "u" instead of v in front of sinθ. Check.


ehild

Yes, I saw that, but it must be a transcription error since it is correct in the later equation.
 
  • #5
A figure would be necessary. I assume that the girl runs to the left so the boy has a horizontal velocity component u to the right. See figure. That component and the boy's radial velocity component v add up and determine the boy's next position.
You see that the horizontal component u makes r increase, but θ decrease.

In case the girl run to the right, the boy would move in the second quadrant, the angle θ would increase as the boy approaches the girl.

What is your set-up? Check the signs.

ehild
 

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  • #6
haruspex said:
If you expand the numerator, the second term is easy to integrate. The first becomes the integral of a cosecant, which is standard enough that you can look it up. If you're interested in doing it from first principles, multiply top and bottom by sine and convert the sin2 in the denominator to 1-cos2. You can then factorise that and expand it using partial fractions. Each fraction is readily integrated.

There is a standard substitution when you have to integrate a rational function of sines and cosines. All trigonometric functions can be written in terms of tan(x/2), and then the substitution z=tan(x/2), dx = 2/(1+z2) dz can be used.

[tex]\sin(x)= \frac {2\tan(x/2)}{1+\tan^2(x/2)}[/tex]
[tex]\cos(x)= \frac {1-\tan^2(x/2)}{1+\tan^2(x/2)}[/tex]

[tex]\int {\frac{1}{\sin(x)}= \frac {1+\tan^2(x/2)}{2\tan(x/2)}dx}= \int {\frac {1+z^2}{2z}2 \frac{1}{1+z^2}dz}=\int {\frac {1}{z} dz}= \log|tan(x/2)|+C[/tex]ehild
 

What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It is commonly used to model physical phenomena in science and engineering.

What is the significance of solving a differential equation?

Solving a differential equation allows us to determine the behavior of a system over time. This is important in understanding and predicting the behavior of many natural and man-made systems.

How do you solve a differential equation?

There are different methods for solving differential equations depending on the type and complexity of the equation. Some common methods include separation of variables, substitution, and using integrating factors.

What is meant by the "Boy & Girl's Meeting Time" in this context?

In this context, "Boy & Girl's Meeting Time" refers to a specific type of differential equation that models the time it takes for a boy and a girl to meet while walking towards each other with different speeds.

Can differential equations be solved analytically or numerically?

Yes, differential equations can be solved analytically using mathematical techniques and formulas, or numerically using computer algorithms and simulations. The approach used depends on the complexity of the equation and the desired level of accuracy.

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