A bullet is fired straight up into the air with an initial speed of 450 m/s. To what maximum height will the bullet reach?

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This question is relatively simple to answer if we ignore air resistance. But actually in this case it will be a very rough estimate, because such a great initial speed means huge air resistance.

I hope this is a sample problem about a free fall and that you don't need to use the result in practice.

The simplest method to find the maximum height, denote it `H_1,` is to consider energy (also denote the initial speed as `V_0`). If we measure potential energy off of the ground level, then at the start a bullet has only kinetic energy `(m*V_0^2)/2.`  At the maximum height, when a bullet starts to fall, the speed will be zero, and a bullet has only potential energy `m*g*H_1.`

Because of energy conservation  `(m*V_0^2)/2=m*g*H_1,`  so

`H_1=(V_0)^2/(2g) approx (450)^2/20` =10125 (m). This is the formal answer. In reality it will be much less, when taking air resistance into consideration.

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