When a projectile is launched horizontally, its horizontal range $R$ is directly proportional to its initial horizontal speed $v_0$ and the time of flight $t$. The formula for the horizontal range is:

$R = v_0 \cdot t$

For a horizontally launched projectile, the time of flight $t$ is determined by the vertical motion, which is governed by the acceleration due to gravity $g$ and the height $h$ from which the projectile is launched:

$t = \sqrt{\frac{2h}{g}}$

Notice that the time of flight $t$ is independent of the horizontal speed $v_0$. Therefore, if the initial speed $v_0$ doubles, the horizontal range $R$ will also double, because $R$ is directly proportional to $v_0$.

So, if the initial speed doubles, the horizontal range will increase by a factor of 2.

Would you like a more detailed explanation or have any questions about this concept?

Here are 5 related questions you might find interesting:

1. How does the launch height affect the horizontal range of a projectile?
2. What happens to the time of flight if the height of the launch is doubled?
3. How does air resistance affect the horizontal range of a projectile?
4. Can you calculate the time of flight for a projectile launched from a certain height?
5. What would happen to the horizontal range if the acceleration due to gravity changed?

Tip: When analyzing projectile motion, always separate the horizontal and vertical components of the motion, as they are independent of each other.