Shortest distance between two bodies? Anyone good at Mechanics, please help...

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Princess Kasumi

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...required here? 1. Distance to be found was d² = 20t² – 4t + 25
The equation was formed from two particles A and B (long question short), the answer for least distance was:
d² (min) = 5 and d (min) = √5.
My question is how? How do I find the least distance, I know the answer but some explanation required how to get it please?

2. At 10:00 a.m, a plane P passes through a point X traveling due south at a speed of 360 km/h. At the same instant a second plane Q is 500 km due west of X and traveling at a speed of 450 km/h towards X. If neither plane changes its speed of direction, find the time, to the nearest minute, when they are closest together.

Thanks for answering.
 
1. Your answer is wrong, or you mistyped the equation.

First you convert it to a function of time. We don't care
that it's d² -- all we care is where is the minimum of it. So we can find minimum of d².

d²(t) = 20t² - 4t + 25

That's a quadratic polynomial with

A=20, B=-4, C=25

Minimum is at the apex of the parabola. The apex, in this case,
is at t = -B/2A = 4/40 = 0.1

You substitute into the equation:
d²(0.1) = 24.8. NOT 5!
Then d=√24.8=4.9800.

2. The simplest way is to express the position in 2D coordinates as a function of time for each body, then calculate distance as a function of time, then find that function's minimum.

Our origin is at X, coordinates are [east, north].

Position of plane P = [0, -360*t], where t is time in hours, and position is in km.
Position of plane Q = [-500+450*t, 0], same thing.
Both P and Q are vectors.

Square of distance is
|| P-Q ||² = d² = (-360t)² + (500-450*t)² =
= 360²*t² + 500² - 2*500*450t + 450²*t²
= 332100 t² - 45000t + 250000

A=332100, B=-45000 C=250000

Apex (minimum/maximum) is at t = -B/2A = 45000/2*332100 = 25/369 h.

60*25/369 = 4 minutes 11 seconds.
 
1. Your answer is wrong, or you mistyped the equation.

First you convert it to a function of time. We don't care
that it's d² -- all we care is where is the minimum of it. So we can find minimum of d².

d²(t) = 20t² - 4t + 25

That's a quadratic polynomial with

A=20, B=-4, C=25

Minimum is at the apex of the parabola. The apex, in this case,
is at t = -B/2A = 4/40 = 0.1

You substitute into the equation:
d²(0.1) = 24.8. NOT 5!
Then d=√24.8=4.9800.

2. The simplest way is to express the position in 2D coordinates as a function of time for each body, then calculate distance as a function of time, then find that function's minimum.

Our origin is at X, coordinates are [east, north].

Position of plane P = [0, -360*t], where t is time in hours, and position is in km.
Position of plane Q = [-500+450*t, 0], same thing.
Both P and Q are vectors.

Square of distance is
|| P-Q ||² = d² = (-360t)² + (500-450*t)² =
= 360²*t² + 500² - 2*500*450t + 450²*t²
= 332100 t² - 45000t + 250000

A=332100, B=-45000 C=250000

Apex (minimum/maximum) is at t = -B/2A = 45000/2*332100 = 25/369 h.

60*25/369 = 4 minutes 11 seconds.
 
1. Your answer is wrong, or you mistyped the equation.

First you convert it to a function of time. We don't care
that it's d² -- all we care is where is the minimum of it. So we can find minimum of d².

d²(t) = 20t² - 4t + 25

That's a quadratic polynomial with

A=20, B=-4, C=25

Minimum is at the apex of the parabola. The apex, in this case,
is at t = -B/2A = 4/40 = 0.1

You substitute into the equation:
d²(0.1) = 24.8. NOT 5!
Then d=√24.8=4.9800.

2. The simplest way is to express the position in 2D coordinates as a function of time for each body, then calculate distance as a function of time, then find that function's minimum.

Our origin is at X, coordinates are [east, north].

Position of plane P = [0, -360*t], where t is time in hours, and position is in km.
Position of plane Q = [-500+450*t, 0], same thing.
Both P and Q are vectors.

Square of distance is
|| P-Q ||² = d² = (-360t)² + (500-450*t)² =
= 360²*t² + 500² - 2*500*450t + 450²*t²
= 332100 t² - 45000t + 250000

A=332100, B=-45000 C=250000

Apex (minimum/maximum) is at t = -B/2A = 45000/2*332100 = 25/369 h.

60*25/369 = 4 minutes 11 seconds.
 
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