Finding closest distance

  • Thread starter thomasrules
  • Start date
In summary: A is moving in the opposite direction of ship B.)The distance between the two ships will be 0.707 km.
  • #1
thomasrules
243
0

Homework Statement


Ship A, sailing due east at 8 km/h, sights ship B 5km to the southeast when ship B is sailing due north at 6km/h. How close to each other will the two ships get?


Homework Equations





The Attempt at a Solution


I drew the picture. Ship a distance is 8t and B is 6t

Use the a^2+b^2=c^2
 
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  • #2
Seems to me there are two coordinates of interest here. Hint.
 
  • #3
sorry I'm lost
 
  • #4
thomasrules said:

Homework Statement


Ship A, sailing due east at 8 km/h, sights ship B 5km to the southeast when ship B is sailing due north at 6km/h. How close to each other will the two ships get?


Homework Equations





The Attempt at a Solution


I drew the picture. Ship a distance is 8t and B is 6t

Use the a^2+b^2=c^2

"Ship a distance is 8t and B is 6t" distance from what? Set up a coordinate system, say with A initially at the origin. What are the coordinates of ship A at time t? What are the initial coordinates of ship B? What are the coordinates of ship A at time t? What is the (square of the) distance between those points as a function of t?
 
  • #5
ok [tex]D^2=(-3.53+6t^2)+8t^2 dD/dt=(200t-42.36)/whatever\\0=200t-42.36\\t=0.212\\[/tex]
I plugged that in the distance formula and got a wrong answer
 
Last edited:
  • #6
Follow HallsOfIvy's suggest and write down the position of the two ships in xy coordinates as a function of time first. It's difficult to tell where you went wrong from what you post.
 
  • #7
yes I have a drawing. I used the sin law to find the distance of the 2 sides.

Coordinates (3.53,0) and (3.53,-3.53)

By the way how do you put spaces in latex coding I tried \\ it didnt work
 
  • #8
Eeh??

Do you even know what coordinates are??

Considered as a function of time, what is ship A's position measured from an origin lying where A was, and sighted B somewhere at t=0?
And, with the same choice of origin, what is B's position as a function of time?
 
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  • #9
At what time? Where is the time dependence? (Not sure about your latex question, sorry).
 
  • #10
yes those are the coordinates.from the origin (0,0) using pythagorean theorem,

i use 5sin45=a=3.53, b=3.53
 
  • #11
1. What choice have you made of positive axes?
2. What was A's position at t=0?
3. What is A's position as a function of time?
4. What is B's position as a function of time?
 
  • #12
A's position at t=0 is (0,0)
A's position as a function of time is (3.53-8t,0)
B's position as a function of time is (3.53,-3.53-6t)
 
  • #13
Is (3.53-8*0,0)=(0,0)?? :confused:?
 
  • #14
ah crap...
A is (8t,0)
 
  • #15
**** or is it
A(8t,0)
B(3.53,-3.53+6t)

My guess is that's right if not I quit school
 
  • #16
You can stay in school! Now what is D^2 as a function of t?
 
  • #17
D^2=(-3.53+6t-0)^2+(3.53-8t)^2

I did derivative. Set it to 0. Got 0.7

Thats not the answer

I'm suppost to plug time into the D^2 and get the distance correct?
 
  • #18
You seem to be trying to do the right thing. What is the derivative of D^2?
 
  • #19
I got it to be:

After the expansion and derivative

dD/dt=200t-98.84/(the rest here doesn't matter cause I'm setting other side to zero)
 
  • #20
All seems ok. What is the answer supposed to be?
 
  • #21
The ships get 0.707km to each other

btw thanks for all your help so far
 
  • #22
You're welcome. But that answer only differs from 0.7 because you are rounding off differently.
 
  • #23
NO! but we only have found the time man not the distance
 
  • #24
thomasrules said:
I got it to be:

After the expansion and derivative

dD/dt=200t-98.84/(the rest here doesn't matter cause I'm setting other side to zero)

You are saying your solution for the TIME is 0.7? That's not what this says.
 
  • #25
OMG YOU ARE CORRECT!

Thanks man. I'm slow. What's your paypal how much do I owe you.
:tongue2:

And now I have another similar one I'm struggling with it :S
 
  • #26
Good luck!
 
  • #27
Because you (cleverly) chose to make A's initial position the origin of your coordinate system, A's position at t= 0 is, indeed, (0,0). Since A is moving due east (and you chose to make the positive x-axis that direction) at 8 km/h, A's position at time t hours is (8t, 0). Yep, that's what you got!

Initially, B was "5 km to the south east" so B's initial position is [itex](5\frac{\sqrt{2}}{2}, 5\frac{\sqrt{2}}{2}[/itex] ([itex]sin(45^o)= \frac{\sqrt{2}}{2})[/itex]. Since B is moving straight North at 6 km/h, B's position at time t hours must be [itex](5\frac{\sqrt{2}}{2}, 5\frac{\sqrt{2}}{2}+ 6t)[/itex]). If [itex]5\frac{\sqrt{2}}{2}= 3.53[/itex](and it is) you are correct. Glad you don't have to quit school!

Now what is the distance between those points? (Hint: since distance is always positive, minimizing distance is the same as mininizing distance squared- so you can ignore the square root in the distance formula.)
 
  • #28
Yea I got it thanks Halls of Ivy...

I get to stay in school :D
 

1. What is the concept of finding closest distance?

Finding closest distance is a mathematical concept that involves determining the shortest distance between two points or objects. It is commonly used in various scientific fields, such as physics, engineering, and geography.

2. How is the closest distance between two points calculated?

The closest distance between two points is calculated using the Pythagorean theorem, which states that the square of the hypotenuse (longest side) of a right triangle is equal to the sum of the squares of the other two sides. This formula allows us to calculate the distance between any two points in a two-dimensional space.

3. What factors affect the closest distance between two objects?

The closest distance between two objects can be affected by various factors, including the initial distance between the objects, their relative speeds and directions of movement, and any external forces acting on them, such as gravity or friction.

4. How is the concept of closest distance used in real-life applications?

The concept of closest distance has many practical applications in fields such as navigation, transportation, and telecommunications. For example, it is used in GPS systems to determine the shortest route between two locations, in traffic engineering to optimize traffic flow, and in satellite communication to ensure accurate transmission of data.

5. Are there any limitations to the concept of finding closest distance?

While the concept of closest distance is a useful tool in many scientific and technological applications, it does have its limitations. For instance, it assumes that the objects being measured are point objects with no size or shape, and it does not take into account any obstacles or barriers that may affect the distance between them.

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