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Both of them will touch the ground simultaneously. The reason is that both of them equal acceleration in the vertically downward direction and also have the same initial velocity in the vertically downward direction.

They will reach the ground at the same time but the horizontal distances will be different because of the different initial velocities in the horizontal direction :)

@Ajinkya Parab
–
Well, Let us consider two bullets B1(With low speed) and B2(With high speed) fired from same location at vertical distance of say 5m above ground level.Let us consider B1 goes 200 meter horizontal distance and falls down to vertical distance of 5 meter. Again consider second bullet B2 travels 2000 meter horizontal distance and falls down to hit the ground to vertical distance of '5 meter + EXTRA'. And if you observe carefully that EXTRA vertical distance is the distance caused due to 'CURVATURE OF EARTH'.
So B2 travels comparatively more vertical distance hence take more time. If any body think this reason
does not support my answer completely I will give another supportive reason in my next reply.

@Ishwar Patil
–
Curvature Of Earth?! In practical problems, we don't consider that! Say, as in your case, it travels 2000m horizontally. This is negligible w.r.t the earth's circumference ( less than a degree). We can safely assume the ground to be horizontal.

@Ajinkya Parab
–
If you observe in very minute calculations(Without neglecting) you see that both bullets do not touch ground the at same time.And my previous example of 2000m path may give negligible difference, but what if you consider higher values.
And if one is trying to think theoretically then take example of a bullet at initial speed
of 2000 km/sec. What you imagine now? The bullet is having high potential and it makes its way
into outer space following a tangential path with the earths spherical surface(If still 2000 km/sec
does not behave that way, then take higher values at which it is possible).And in this example
the bullet will never hit the ground since it will have evaded the earth gravity.
However if one tries to see the practical stuffs then there is no need to worry about which
bullet falls first(Of course who wants to know about it in practical scenarios). And still if
some one is interested to perform it practically then the parameters in question are insufficient.
You need to have lot of practical considerations like land topography,wind speed & directions..etc and
which in turn will prove that both the bullets do not touch ground simultaneously and are dependent on many PRACTICAL parameters.
So, in both ways practical and theoretical ways both bullets do not touch ground simultaneously.

@Ishwar Patil
–
What I mean by PRACTICAL is, 'under IDEAL conditions (neglecting air resistance etc etc :p ) but POSSIBLE practically'! Speeds of 2000m/s or higher are not practically achievable by real life bullets!! The highest speed being 4000ft/s i.e.approx 1200m/s {googled :p}
So PRACTICALLY, you CAN consider the ground flat! And you SHOULD !! ( unless you intend to calculate a difference of microseconds or even lesser, and use complicated calculus as you will also consider variations in acceleration due to gravity with height, aiming to be so precise! :p )

@Ajinkya Parab
–
I want to point out two things in the reply.
1. What is reason for not considering earths curvature if bullet travels initially at
1200 m/s. There is no proof mentioned why one SHOULD NOT consider curvature.
2. In my opinion making use of calculus for height variation for precision is not required in this case.
(It may be or may not be required,you may think I am wrong in this point because discussing
on it may take more deviation from topic)
For every 1 mile on earth there will be depression in ground level by 8 inches(~20 cm approx).
So imagine if you fire bullet at height of 1 meter (100 cm) above ground level and it travels
to a distance of 1.5 mile, your extra vertical distance due to curvature will be 30 cm.
So for actual 100 cm vertical distance that you initially fired, the bullet is travelling 130 cm
vertically downward. There is 30% increase in vertical distance. This is large percentage
difference in distance and hence the time difference will also be large relatively.
And most of all it is left to the observer on what precision level he require the answer to be.
Because the time taken by bullet to hit the ground itself is too small and if he is worried about such
small quantity then the curvature of earth which is also too small need to be considered.

@Ishwar Patil
–
Ok.. So tell me this.. 2 bullets are fired simultaneously, horizontally from the top of a building 10m high with speeds 800 m/s and 1000m/s respectively. So what time will each of them take to reach the ground ( with as much precision as you wish ) ?
(With all the assumptions I made, if g=10m/(s×s), both bullets will take 1.414s.)

@Ajinkya Parab
–
Ok from discussion we can deduce that when initial velocities differ by large value then
the time difference in touching ground will be LARGE(showed example of 30% difference in my
previous reply). Similarly if you take examples of bullets with not much velocity difference
(Like 800 m/s and 1000 m/s) the time difference will be very minute or say VERY NEGLIGIBLE(It is
ok even if you completely IGNORE difference).
So ultimately it is the EXAMPLE that you consider and the amount of ACCURACY you want.

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This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

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## Comments

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TopNewestBoth of them will touch the ground simultaneously. The reason is that both of them equal acceleration in the vertically downward direction and also have the same initial velocity in the vertically downward direction.

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I think the question is stated for HORIZONTALLY fired up bullets. You are missing that point in first place.

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They will reach the ground at the same time but the horizontal distances will be different because of the different initial velocities in the horizontal direction :)

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For every 1 mile on earth there will be depression in ground level by 8 inches(~20 cm approx). So imagine if you fire bullet at height of 1 meter (100 cm) above ground level and it travels to a distance of 1.5 mile, your extra vertical distance due to curvature will be 30 cm. So for actual 100 cm vertical distance that you initially fired, the bullet is travelling 130 cm vertically downward. There is 30% increase in vertical distance. This is large percentage difference in distance and hence the time difference will also be large relatively. And most of all it is left to the observer on what precision level he require the answer to be. Because the time taken by bullet to hit the ground itself is too small and if he is worried about such small quantity then the curvature of earth which is also too small need to be considered.

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The bullet with low initial velocity will hit ground first..

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