Mathematics VI: speed

[⛄]

Suppose a car has driven along an entire ninety-mile-long street at an average speed of 60 mph.
At the beginning of the street the vehicle was driving at an even speed of 80 mph.
At a certain point it decelerated to an even speed of 40 mph.
After how many miles did the car slow down?
(Brake paths are neglectable.)

[H.E. VOLODYMYR ZELENKSYY]

Since it started at 80 mph and ended at 40 mph, it must have been decelerating the entire time to have an average speed of 60 mph, no? That's just my intuition, but I'd have to do the math later.

[⛄]

Since it started at 80 mph and ended at 40 mph, it must have been decelerating the entire time to have an average speed of 60 mph, no? That's just my intuition, but I'd have to do the math later.

No. The car drove both parts of the street at an even speed.
Imagine it decelerated at one point from 80 mph to 40 mph within 0 seconds. I don't wanna make it too complicated.

[H.E. VOLODYMYR ZELENKSYY]

Since it started at 80 mph and ended at 40 mph, it must have been decelerating the entire time to have an average speed of 60 mph, no? That's just my intuition, but I'd have to do the math later.

No. The car drove both parts of the street at an even speed.
Imagine it decelerated at one point from 80 mph to 40 mph within 0 seconds. I don't wanna make it too complicated.

0 seconds? It'd have to decelerate 45 miles along the street, then, no?

[⛄]

Since it started at 80 mph and ended at 40 mph, it must have been decelerating the entire time to have an average speed of 60 mph, no? That's just my intuition, but I'd have to do the math later.

No. The car drove both parts of the street at an even speed.
Imagine it decelerated at one point from 80 mph to 40 mph within 0 seconds. I don't wanna make it too complicated.

0 seconds? It'd have to decelerate 45 miles along the street, then, no?

80 mph - hard brake - 40 mph

Come on, you understand the question.

[MaxQue]

Suppose a car has driven along an entire ninety-mile-long street at an average speed of 60 mph.
At the beginning of the street the vehicle was driving at an even speed of 80 mph.
At a certain point it decelerated to an even speed of 40 mph.
After how many miles did the car slow down?
(Brake paths are neglectable.)

Well, it's going at 80 mph half for half the path and 40 mph for half the path.

So, it slowed down at 45 miles.

But I'm not sure I'm seeing the point of that question, it's pretty simple.

[True Federalist (진정한 연방 주의자)]

Suppose a car has driven along an entire ninety-mile-long street at an average speed of 60 mph.
At the beginning of the street the vehicle was driving at an even speed of 80 mph.
At a certain point it decelerated to an even speed of 40 mph.
After how many miles did the car slow down?
(Brake paths are neglectable.)

Well, it's going at 80 mph half for half the path and 40 mph for half the path.

So, it slowed down at 45 miles.

But I'm not sure I'm seeing the point of that question, it's pretty simple.

Not quite.

It's a system of two equations, one for the distance and one for time.

Let x be the number off miles driven at 80mph and y be the number at 40mph

x + y = 90 is the equation for the 90 miles
3/4 x + 3/2 y = 90 is the equation for the 90 minutes.

Solve it and you get x=60 and y=30, so it slows down at 60 miles.

[MaxQue]

But from where 3/4 and 3/2 are appearing?

[H.E. VOLODYMYR ZELENKSYY]

Since it started at 80 mph and ended at 40 mph, it must have been decelerating the entire time to have an average speed of 60 mph, no? That's just my intuition, but I'd have to do the math later.

No. The car drove both parts of the street at an even speed.
Imagine it decelerated at one point from 80 mph to 40 mph within 0 seconds. I don't wanna make it too complicated.

0 seconds? It'd have to decelerate 45 miles along the street, then, no?

80 mph - hard brake - 40 mph

Come on, you understand the question.

I answered it. (45*80 + 45*40)/90 = 60, no?

[Foucaulf]

My preferred method involves only one equation:

t = t_1 + t_2, where t is total time traversed, t_1 is total time traversed at 80 mph and t_2 is total time traversed at 40 mph.

Plugging in the formula d = vt --> t=d/v yields

90/60 = x/80 + (90-x)/40; the x is the distance travelled at 80 mph, what we really want.

Solve the equation to obtain

120 = x + 2(90-x) -> x = 60. Then t_1 = 3/4, t_2 = 3/2.

It's a standard tricky algebra question. The same principle applies as for a question like "if a farmer had two chickens worth $110 and one is worth $100 more than the other what is its value" - instead of assuming some value outright, set a variable and list out conditions you do know.

EDIT: How about a harder variant:

At some point the car starts to slow down at a rate of 60 mph per minute until it reaches 40 mph. After how many miles did the car start to slow down? What if it's 100 mph, 120 mph or 200 mph? Do your answers approach to anything?

[True Federalist (진정한 연방 주의자)]

But from where 3/4 and 3/2 are appearing?

At 80mph, a mile is traveled in 3/4 of a minute.
At 40mph, a mile is traveled in 3/2 of a minute.

[Chancellor Tanterterg]

My preferred method involves only one equation:

t = t_1 + t_2, where t is total time traversed, t_1 is total time traversed at 80 mph and t_2 is total time traversed at 40 mph.

Plugging in the formula d = vt --> t=d/v yields

90/60 = x/80 + (90-x)/40; the x is the distance travelled at 80 mph, what we really want.

Solve the equation to obtain

120 = x + 2(90-x) -> x = 60. Then t_1 = 3/4, t_2 = 3/2.

It's a standard tricky algebra question. The same principle applies as for a question like "if a farmer had two chickens worth $110 and one is worth $100 more than the other what is its value" - instead of assuming some value outright, set a variable and list out conditions you do know.

EDIT: How about a harder variant:

At some point the car starts to slow down at a rate of 60 mph per minute until it reaches 40 mph. After how many miles did the car start to slow down? What if it's 100 mph, 120 mph or 200 mph? Do your answers approach to anything?

42 is the answer to all things.

[Smid]

Since it started at 80 mph and ended at 40 mph, it must have been decelerating the entire time to have an average speed of 60 mph, no? That's just my intuition, but I'd have to do the math later.

No. The car drove both parts of the street at an even speed.
Imagine it decelerated at one point from 80 mph to 40 mph within 0 seconds. I don't wanna make it too complicated.

0 seconds? It'd have to decelerate 45 miles along the street, then, no?

80 mph - hard brake - 40 mph

Come on, you understand the question.

But it only applies to spherical chickens in a vacuum...

[⛄]

EDIT: How about a harder variant:

At some point the car starts to slow down at a rate of 60 mph per minute until it reaches 40 mph. After how many miles did the car start to slow down? What if it's 100 mph, 120 mph or 200 mph? Do your answers approach to anything?

Here's my answer:

You'll build a system of linear equations:

(I)  80 mph * t₁ + ⅔ mi + 40 mph * t₂ = 90 mi

(II) t₁ + 1/90 h + t₂ = 1.5 h


⇒ t₁ = t₂ = 67/90 h = 2,680 s

⇒ s₁ = 59 mi 2933.3 ft

Is that correct?