Which Bullet Will Run Out Of Breath First?

Classical Mechanics Level 1

Two bullets of equal masses are fired horizontally with speeds of 10 m/s and 20 m/s from the same height at the same time. Which bullet will hit the ground first?

Neglect air resistance, the curvature of the earth, lift effects on the bullet, etc.

The faster one will hit the ground first The slower one will hit the ground first Both will hit the ground simultaneously Depends on the mass of the bullets

Rohit Gupta by Rohit Gupta , 34, India

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7 solutions

Discussions for this problem are now closed

Nihar Mahajan
Jan 16, 2016

Though the magnitude of horizontal component of velocity is varying , their magnitude of vertical component of velocity increases at the same rate since the gravitational acceleration is constant for both the bodies and that is why they would reach the ground at same time (but at different positions). (Note we are ignoring air resistance).

82 Helpful 2 Interesting 1 Brilliant 1 Confused

Extreme example: 1st bullet a 1 m/s 2nd bullet and 200,000,000 m/s. The second bullet would never hit the ground.

Brian Wang - 5 years, 5 months ago

No. Increased muzzle velocity results in an increase in spin applied by the rifling. This spin generates lift similar to an airplane and will also create a gyroscopic stabilizing effect (up to a certain velocity). A low velocity bullet will lack stability and "tumble" sooner, losing the effect of lift. This solution is incorrect.

Chad Wallace - 5 years, 4 months ago

No air resistance, so neither bullet should "tumble"

Brian Wang - 5 years, 4 months ago

@Brian Wang – If you were to say that the bullets were fired in a vacuum I would agree that both bullets hit the ground at the same time assuming a perfectly flat plane. The lift generated by the Magnus effect from the rifling (spin) while certainly generated by the fluid in which the bullet is fired is in itself not air "resistance" and has a very measurable effect on air time. Tumbling results in a loss of this effect, again separate and distinct from air resistance.

Chad Wallace - 5 years, 4 months ago

Haha , but I don't think a mechanism is available yet , which can fire something that faster :P

Nihar Mahajan - 5 years, 5 months ago

Alright, suppose 1 bullet is fired at 10 m/s, and another bullet is fired at 0 m/s. The bullet fired slower will never leave the barrel, and thus never hit the ground.

Brian Wang - 5 years, 5 months ago

@Brian Wang – A bullet is fired means it has released with some non-zero velocity.

Akhil Bansal - 5 years, 5 months ago

@Akhil Bansal – alright, 0.0000000000000000001 m/s

Brian Wang - 5 years, 5 months ago

@Brian Wang – Do you really think we can fire a particle with such negligible velocity.?

Akhil Bansal - 5 years, 5 months ago

@Akhil Bansal – sure, just massively increase the mass of the projectile and decrease the propellent, i.e blow on the bullet.

Brian Wang - 5 years, 5 months ago

@Brian Wang – There should actually be two cases in this question. 1.) When both bullets are fired at different normal speeds. 2.) When one bullet is just fired and the other bullet is fired extremely fast. I guess the question is not clear a bit.

Gokul RejithKumar - 5 years, 5 months ago

Sure. Anything 'fired' with a very high speedy (like 20.000km/h) will fly away from earth.

Mauricio Braga - 5 years, 4 months ago

That's where your logic is flawed, at first thought it'd seem like the second bullet would forever be in linear motion for a long distance, a long distance yes, it might even circle the earth twice (JK), but it's travelling at such a great speed it'll only be so quick for it to reach the ground. Many tried to respond that those speeds for practically impossible, but even if they were, under given conditions and neglecting air resistance, they'd always reach the ground at the very same instant.

Pradeep Daniel - 5 years, 4 months ago

200,000,000 m/s is well over the escape velocity of earth from the surface. Light travels at ~300,000,000 m/s. The escape velocity from earth's surface is ~12,000 m/s. So the bullet will not orbit Earth, it would escape the solar system (assuming it's pointing the right way)

Brian Wang - 5 years, 4 months ago

@Brian Wang – Except, an escape velocity is irrelevant when the earth's curvature is neglected(as mentioned in the question), that is assuming that it's an infinite flat plane. Now the conditions are quite restricted to apply real world logic, but apart from this very specific question, you are obviously correct.

Pradeep Daniel - 5 years, 4 months ago

Really bro ??? Really ? 200000000 m/s.........

Vinod Kumawat - 5 years, 4 months ago

Why is it not related to the mass?

Arjunesh Namboothiri - 5 years, 4 months ago

"their vertical component of velocity increases at the same rate."

Vectors can't "increase" or "decrease". Change your wording.

Venkata Karthik Bandaru - 5 years, 4 months ago

Sorry I meant magnitudes. Thanks!

Nihar Mahajan - 5 years, 4 months ago

I'm confused. I'm firing one 9mm round, and one 50mm round. Why does the heavier (ok, greater mass) one not land first?

Mam Bach - 5 years, 4 months ago

you are neglecting momentum fast speed means more momentum slow speed means less momentum so bullet with fast speed will hit the ground when it will lose all it's momentum and I think bullet with less speed will loose it's momentum faster than bullet with more speed.

Hamza Arshad - 5 years, 4 months ago

If you throw a feather and a stone in horizontal direction from a high cliff (neglect the air resistance) , which do you think will reach the ground first? Or both would reach at the same time?

Nihar Mahajan - 5 years, 4 months ago

Can we use the idea of projectile motion here?

Sridhar Sri - 5 years, 4 months ago

It is kind of counterintuitive, but this means that no mater how fast the bullet travels, it would hit the ground as fast as a free falling object because its vertical velocity components are the same. It is important to bear in mind the fact that no other component is included, like the interaction with wind, which makes it counterintuitive, as pure math may result many times.

Alfredo Alvarez - 5 years, 4 months ago

Still your ans is not understood.

ankit raj - 5 years, 4 months ago

I am confused...no where does it say that the projectiles are identical; if F=ma then shouldn't the force of the earth applied to the projectiles affect the projectile with greater mass more than the one with less mass?

Sergio Lopez - 5 years, 4 months ago

There is no involvement of forces, it is just simple kinematics.

Venkata Karthik Bandaru - 5 years, 4 months ago

Yes, the force applied on the greater mass projectile will be higher, that is why the mass was neglected

Jehad Alhajji - 5 years, 4 months ago

I am also confused from the answer. The mechanical energy (ME) in the "bullet" system is constant (i.e. Kinetic Energy (KE) + Potential Energy(PE) ). The first Bullet's ME on ground equals the KE (PE is zero) and equals 1/2 x m1 x v1^2 (where m1 and v1 are the mass and velocity of the first bullet). At peak height, when the bullet stop ascending and starts falling, its ME will be equal to its potential energy i.e. m1 x g x h1 (where h1 is the height the first bullet will reach). solving both for h1 gives h1 = v1^2 / (2 x g). Hence it is clear that the height the bullet will reach is a function of its velocity.

Now the time taken to fall from a height h1 under gravity (ignoring air resistance) equals sqrt ( 2 x h / g). Hence solving to get the time for the first bullet, t1 = v1 / g

If the first bullet fired is faster than the second, irrespective of its weight, it will reach a higher height and will take longer to fall. Hence I am not in full agreement with the proposed answer.

Kind regards

Mohamed Eltawil - 5 years, 4 months ago
Arjen Vreugdenhil
Jan 27, 2016

I did assume the ground was level. On a sloping hill, the slower one would hit the ground first :)

34 Helpful 0 Interesting 0 Brilliant 0 Confused

That depends on which direction the ground was sloping...

Nate Tucker - 5 years, 4 months ago
TwixOps .
Jan 19, 2016

One would think that general relativity would come into play in such a problem.

Extreme example. Imagine two bullets are fitted above an endless plane, one at 10m/s and the other at .9c. From the relative perspective of each bullet, they took exactly a second to hit the floor, yet in actuality the high speed bullet would experience time dilation; meaning a stationary observer would observe it take 2.29s to fall. The slower round on the other hand would take only something like 1.0000001s to land.

At more everydayspeeds such as those a bullet could be fired at, the effect would be immeasurably small, but still present, unless my assumptions are wrong

30 Helpful 0 Interesting 0 Brilliant 0 Confused
Akshay Sharma
Jan 27, 2016

The horizontal and vertical components are independent of each other.

6 Helpful 0 Interesting 0 Brilliant 0 Confused
Achille 'Gilles'
Jan 19, 2016

Initial vertical speed are the same so both bullet will gain vertical speed at the same rate and touch ground at the same time.

4 Helpful 0 Interesting 0 Brilliant 0 Confused

What about centrifugal force?

Cebanu Iulian - 4 years, 11 months ago

Akin to Galileo's thought experiment about the two balls dropped from the Tower of Pisa. The key in bnoth instances is that Mass always equals Mass: that is, inertial mass = gravitic mass.

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Shivam Patil
Jan 17, 2016

If all the external factors are neglected, irrespective of the horizontal component of velocity, the gravitational acceleration of both the bullets is the same and since the bullets are fired in the horizontal direction, there's no velocity/acceleration is the vertical direction. Hence if fired at same heights, both will reach the ground simultaneously.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

What if one reaches orbit? Or of one doesn't exit the barrel?

Brian Wang - 5 years, 4 months ago

That's an another case. I mentioned if all the external cases are neglected. To be precise, what I said is true if you do this experiment on a flat surface of infinite length with even distribution of atmosphere and gravitational force.

Shivam Patil - 5 years, 4 months ago

But that's not the question. the question simply says "the same place"

Brian Wang - 5 years, 4 months ago

@Brian Wang – BBut the time of flight is different ??? Time is proportional to velocity

Debasis Rath - 5 years, 4 months ago

@Brian Wang – So ? It can even be the place I'm talking about. Here, they have mentioned same place so as to make sure that the gravitational acceleration is same.

Shivam Patil - 5 years, 4 months ago

a bullet flies up and then down, crossing the line of sight twice b4 impact on target. there is acceleration in the vertical direction.. and it depends on the mass of the bullets AND their respective speed to determine which will hit the ground first

Jesse Johns - 5 years, 4 months ago

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