Can the Trajectory of a Bullet Be Modeled with a Single Mathematical Function?

[lss1]

I've been mulling this over all weekend, and I've decided to get some help on this. The problem is writing a function to describe a bullet's path. I've asked two people about it my Physics teacher (who said he didn't know how) and my French teacher, who was a nuclear engineer for the US Navy (who said it was impossible). I don't know much about ballistics, but I am very willing to learn.

My Physics teacher started out with the equation $$y = v_y t + \frac{1}{2} a t^2$$ and the equation $$x = v_x t.$$ So I've been looking for a way to combine these two functions. I asked my French teacher about it and he said it was impossible because at the beginning of the travel-path, the motion is dominated by the x-component, and as it goes on the velocity in the x-direction slows down, and the y-acceleration becomes more dominant. He said that as the motion changes from x-dominated to y-dominated, the variable, t, becomes two different variables, and therefore cannot be written in the same function. I've been thinking it could work as a multivariable function, but I'm not sure.

Any help would be gratefully appreciated.

 

[berkeman]

I've been mulling this over all weekend, and I've decided to get some help on this. The problem is writing a function to describe a bullet's path. I've asked two people about it my Physics teacher (who said he didn't know how) and my French teacher, who was a nuclear engineer for the US Navy (who said it was impossible). I don't know much about ballistics, but I am very willing to learn.

My Physics teacher started out with the equation $$y = v_y t + \frac{1}{2} a t^2$$ and the equation $$x = v_x t.$$ So I've been looking for a way to combine these two functions. I asked my French teacher about it and he said it was impossible because at the beginning of the travel-path, the motion is dominated by the x-component, and as it goes on the velocity in the x-direction slows down, and the y-acceleration becomes more dominant. He said that as the motion changes from x-dominated to y-dominated, the variable, t, becomes two different variables, and therefore cannot be written in the same function. I've been thinking it could work as a multivariable function, but I'm not sure.

Any help would be gratefully appreciated.

Welcome to the PF.

Projectile motion is pretty simple to deal with until you introduce air resistance. This wikipedia page should get you going:

http://en.wikipedia.org/wiki/Projectile_motion

Are you wanting to incorporate the effects of air resistance at some point?

 

[SteamKing]

Lesson: don't ask a French teacher to do the Physics teacher's job.

If you shoot a projectile with an initial velocity v at an angle theta to the horizon, the velocity can be decomposed into a horizontal component (vx) and a vertical component (vy). The time variable t is the same. Assuming there is no resistance to the motion of the projectile, the horizontal velocity is not diminished, while the effect of gravity acts to diminish the vertical velocity. At some time after launching the projectile, the vertical velocity drops to zero (while the horizontal velocity is undiminished), and the projectile begins to drop toward the ground (in other words, the vertical velocity has changed sign).

While you can't combine the expressions for the displacements x and y into a single equation, you can compute values of x and y at a single time t. The values of x and y can be thought of as components of a position vector which locates the projectile relative to where it was initially fired.

 

[VantagePoint72]

While you can't combine the expressions for the displacements x and y into a single equation, you can compute values of x and y at a single time t. The values of x and y can be thought of as components of a position vector which locates the projectile relative to where it was initially fired.

Unless by "combine these two functions" OP meant "write y as a function of x". If that's the case, OP: that's also easy, and is explained under "Parabolic trajectory" in berkeman's Wikipedia link.

 

[milesyoung]

Couldn't the OP use the second equation to eliminate time from the first? Assuming he/she is just looking for an equation to relate the x and y coordinates of the bullet trajectory.

Edit:
Merde!