To figure out how far a projectile will go using Newton's Laws, we need to understand a few basic things about its movement. Here’s a simple way to think about it:

1. Identify Initial Speed:

   First, we need to know the speed at which the projectile is launched. We call this speed ( v_0 ).

   We can break it into two parts:

   • Horizontal speed (( v_{0x} ))
   • Vertical speed (( v_{0y} ))

   We can find these two speeds using some basic math:

   • ( v_{0x} = v_0 \cos(\theta) )
   • ( v_{0y} = v_0 \sin(\theta) )

   Here, ( \theta ) is the angle at which the projectile is launched.

2. Calculate Time in the Air:

   Next, we want to know how long the projectile will stay in the air. This is called the flight time.

   We can figure this out using the vertical motion and gravity. The time ( t ) it takes for the projectile to go up, reach its highest point, and then come back down is calculated like this:

   • ( t = \frac{2v_{0y}}{g} )

   In this formula, ( g ) is the acceleration due to gravity, which is about ( 9.81 , m/s^2 ).

3. Calculate the Range:

   Finally, we can find out how far the projectile travels horizontally. This distance is called the range, ( R ).

   To calculate the range, we multiply the horizontal speed by the total time the projectile is in the air:

   • ( R = v_{0x} \times t )

By following these steps, it becomes easier to see how far a projectile will travel. This method also fits well with Newton's Laws of motion and forces!