Solving the Problem: A Bullet is Fired Horizontally Aiming at an Object

[wirefree]

Question: A bullet is fired horizontally aiming at an object which starts falling at the instant the bullet is fired. Show that the bullet will hit the object.


Problem: The conventional way students are advised to address this question is to consider that all limits are bounded: the horizontal distance between the object and the bullet is not too large, the bullet is shot with sufficient speed. The solution, then, simply requires considering the vertical component of the bullet's initial velocity and calculating the distance covered by it in a given time 't', which would be, in fact, equal to that covered by the object in the same time. This concludes the solution.

But, surely, a more rigorous treatment is required; nothing in the question suggests that the setup is perfectly designed.


Approach: Prove that the time taken by the bullet to cover the vertical distance to the horizontal projection of the point where it meets the object is equal to the time taken by the object to cover that same vertical distance is equal to the time taken by the bullet traveling at its horizontal component of velocity to cover the horizontal distance between the gun and the object.

I'll share by attempt in the next post.

At this point, I would appreciate some thoughts on how everyone sees the situation and chooses to address the question.



wirefree

 

[HallsofIvy]

The simplest way to look at it is just what Galileo showed- all objects fall at the same rate- with the same acceleration. Break this into horizontal and vertical components.

 

[Doc Al]

The original solution seems perfectly rigorous (and simple!) to me. Just realize that the bullet and object are always at the same height. So as long as the bullet has enough horizontal speed, it will hit its target.

(A variation of this problem is to have the object at some arbitrary height, with the bullet aimed directly at it. Will the bullet hit its target then?)

 

[A.T.]

nothing in the question suggests that the setup is perfectly designed.

Which design "flaw" isn't covered by the original solution?

 

[SteamKing]

IMO, all textbook problems are idealized to a certain extent. The real world is too messy a place to jump into without some preparation.

If the OP thinks the fired bullet problem he describes is too idealized, he should glom onto a textbook in exterior ballistics, especially one dealing with long-range firing solutions.

 

[wirefree]

Thanks all.

I see my mistake.

Regards,
wirefree

 

[CWatters]

This part of your approach seem unnecessary..

Prove that the time taken by the bullet to cover the vertical distance <snip>... is equal to the time taken by the bullet traveling at its horizontal component of velocity to cover the horizontal distance ..<snip>