Question

Equation derivation

Answer 1

The horizontal distance depends on the horizontal speed, while the vertical distance depends on the vertical speed. Let's start with the horizontal distance.

For the horizontal distance, the horizontal speed stays the same the entire time because there is no force acting in the horizontal direction and therefore no horizontal acceleration. This means the horizontal distance travelled in time t is given by:

\[d_{H}=vt\]
For the vertical distance, the vertical speed starts at 0 (the bullet is fired horizontally with no vertical speed component), but this speed increases in the downward direction due to the force of gravity. The speed increases at a rate equal to g, the acceleration due to gravity. Now, the distance traveled is given by the following equation:

\[d=v _{0}t+\frac{ 1 }{2 }at^2\]

We know already that the starting speed, v(o), is 0. We also know that the acceleration here is g. This means we can rewrite the above equation for vertical distance as:

\[d_{V}=\frac{ 1 }{2 }gt^2\]

The question asks for the ratio of horizontal distance to vertical distance, so let's combine the two equations we came up with:

\[\frac{ d_{H} }{ d_{V} }=\frac{ vt }{ \frac{ 1 }{ 2 }g t^2}=\frac{ 2v }{ g t }\]

Answer 2

g here u are taking as acceleration v/t?

Answer 3

oh wait lol i didn't notice the going up part

Answer 4

THANK YOU like really soooooo much! its perfectly clear :D

Answer 5

really appreciated the long explanation, much obliged.

Answer 6

Not a problem - glad it was helpful!