Projectile Motion: Solving for Meeting Point of Projectile and Falling Object

[yoniwei]

Homework Statement

Prove mathematically that regardless of the range and initial speed, a projectile and a falling object will always meet.

Homework Equations

Vf^2 = Vi^2 + 2ad, d = vi t + 1/2 at^2, d = vt.. etc basic constant velocity and acceleration formulas.

The Attempt at a Solution

this question followed one with specific information given (which I was able to easily solve) and I know how to explain the answer in words; a projectile is under the same force of gravity as th e falling object, so after a certain amount of time the horizontal component of speed will meet up with the falling object, obviously if there is less distance and/or a higher horizontal speed this will occur quicker, but how do I show this conclusion using mathematical formulas?

 

[LowlyPion]

If you are in the frame of reference of the falling object, and it is aimed at you and is fired at the moment you drop, then it will strike you straight away, because you and the projectile are moving together. From your point of view it will be traveling straight at you the whole way.

 

[yoniwei]

If you are in the frame of reference of the falling object, and it is aimed at you and is fired at the moment you drop, then it will strike you straight away, because you and the projectile are moving together. From your point of view it will be traveling straight at you the whole way.

I understand the concept in words but I can't come up with a mathematical equation to prove it. Do I just draw a graph to show that the vertical components are the same and that the amount of times it takes to reach the object is just the horizontal component calculated via d = vt?

 

[LowlyPion]

You can note that in the accelerating frame of reference of the object/projectile world there is no net acceleration on either relative to the other.

You know the time to fall is given by Height = 1/2*g*t²
And the distance is D

So long as the projectile has a V greater than the D/t then it should strike before the ground intervenes to interrupt things.