[Math] A car is moving at 40 km/h. A fly at 100 km/h, starts from wall towards the car(20 km away)flies to car and back. How many trips can it make

algebra-precalculus

A car is moving at a constant speed 40 km/h along a straight road which heads towards a wall.A fly flying at a constant speed of 100 km/h, starts from wall the towards the car at a instant when the car is 20 km away, flies until it reaches the car and comes back to the wall at the same speed.It continues to fly between the car and the wall till the car reach the wall. How many trips has it made between the car and the wall?

I don't expect a brute force because I already did that. Some thing like arithmetic/geometric/harmonic progression will satisfy my curiosity.

Best Answer

It has made infinitely many trips. Every trip will be shorter than the last, but the fly will always reach the wall before the car, so it will always have room for one more trip. And one more. And one more.

The total length the fly flies is 50km, as the car crashes into the wall exactly 30 minutes after the whole experiment started.

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Naturally, formalizing the question in a post helped me solve it. Here's what I came up with for posterity's sake:

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Where Vm is highest attained speed, ta, td, ka and kd are the time and k constants for acceleration and deceleration, and d is the distance to travel.

PS. If someone were kind enough to edit this stuff to use the fancy notation that would be great.

Best Answer

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