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I was trying to prove that horizontal velocity of a projectile remains constant mathematically, but can't fully come close to proving it. Is it possible to do it, or do we have to live with this assumption which is proven correct only experimentally. |
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You certainly cannot prove it mathematically because it is called the Netwon's first law, which is proved by experiment. It states that an object with no external force will move with constant velocity. Since $x$ and $y$ component are independent, so the horizontal velocity is certainly constant (when there is no friction). |
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I'm guessing you're looking at a projectile or something similar where the force is acting only in a vertical direction. If so the proof is very simple. The infinitesimal change in velocity is related to the acceleration by: $$ \vec{dv} = \vec{a} dt $$ If we write the vectors out as horizontal and vertical components we get two equations: $$ \vec{dv_x} = \vec{a_x} dt $$ $$ \vec{dv_y} = \vec{a_y} dt $$ And $\vec{a_x} = \vec{F_x}/m $ where $\vec{F_x}$ is the horizontal component of the force and $m$ is the projectile mass. Since the gravitational force is vertical $\vec{F_x} = 0$ and therefore $\vec{a_x} = 0$ and therefore $ \vec{dv_x} = 0$ i.e. the change in the horizontal velocity is zero. But this is such a simple argument I wonder if I have misinterpreted your question. If so add a comment or edit your question and I'll try to respond. |
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