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I am new to physics.

I’m confused about how an bullet shot horizontally would land at the same time with as a bullet dropped vertically (ignoring air resistance and the curvature of Earth).

Having just learnt about inertia previously, I am confused: Why wouldn't it take time for gravity to overcome the horizontal inertia of the bullet?

Maybe I am confusing momentum with inertia? Does inertia increase with velocity? (I haven't learnt that yet)

EDIT: I realize my mistake now. I saw a video on inertia that explained how cars must first overcome their "forwards inertia" in order to turn. This misled me into believing horizontal/vertical inertia were dependent.

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If you shoot a bullet horizontally with the velocity $v_x$, the gravitational force acts perpendicular to the initial trajectory of the bullet. Since you can decompose a particle's motion (in particular its velocity) into its orthogonal components, you have a component $v_x$ for the horizontal motion and a component $v_y$ for its vertical motion (i.e. towards the ground). The gravitational force leads to an acceleration of the bullet towards the ground, and the acceleration is always in the same direction as the force. Therefore, gravitation will only change the vertical speed component $v_y$. More important for your question, gravitation does not "care" whether there is an initial horizontal velocity. $\vec{F} = \begin{pmatrix} 0 \\ -mg \end{pmatrix} = m\cdot \vec{a}$.

Note that only the y component of $\vec{a}$ is non-vanishing. If you want to calculate the time it takes for the bullet to hit the ground, you need the formula $s_y = \frac{1}{2} a_y t^2$, assuming that no initial velocity in the direction of $y$ is given. In the direction of $x$, the bullet is not accelerated, i.e. $a_x = 0$. Solving for $t$ leads to: $t = \sqrt{\frac{2s}{a_y}}$. Hence, the time is independent from $v_x$.

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Inertia does not have a direction in linear motion. Inertia is the "resistance" of a body to a change in its state of motion. It measured by the mass of the body, which does not increase with velocity (at least not in Newtonian physics) and clearly does not depend on the direction of motion: a body does not resist "more" if you try to push it left or right.

To answer your question: there is no such thing as "horizontal inertia" for a bullet.

For completeness and in contradistinction with linear motion, the situation for rotational motion is different. In rotational motion there is a moment of inertia which depends on the mass distribution about the axis of rotation and the direction of this axis rotation: rotating a cylinder about its symmetry axis is easier than rotating it about an axis perpendicular to the axis of the cylinder.

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