To determine the maximum range of a bullet fired from the bottom of an inclined plane, we need to analyze the projectile motion involved. The inclined plane makes an angle β with the horizontal, and the bullet is fired with an initial velocity r. The goal is to find the range of the bullet along the inclined surface.
Understanding the Components of Motion
When a projectile is fired at an angle, its motion can be broken down into two components: horizontal and vertical. For our inclined plane scenario, we need to adjust these components based on the angle of inclination.
Breaking Down the Initial Velocity
The initial velocity r can be resolved into two components relative to the inclined plane:
- Perpendicular Component: This is the component acting against gravity, calculated as r sin(β).
- Parallel Component: This is the component acting along the inclined plane, calculated as r cos(β).
Equations of Motion
To find the maximum range, we need to apply the equations of motion. The time of flight (T) can be derived from the vertical motion. The bullet will rise and then fall back down to the inclined plane. The vertical motion can be described by the equation:
y = r sin(β) T - (1/2) g T²
Where y is the vertical displacement, g is the acceleration due to gravity, and T is the time of flight. Since the bullet will return to the inclined plane, we set y equal to zero:
0 = r sin(β) T - (1/2) g T²
This simplifies to:
T (r sin(β) - (1/2) g T) = 0
From this, we find that T = 0 (initial launch) or:
T = (2 r sin(β)) / g
Calculating the Range
Next, we can find the range (R) along the inclined plane. The horizontal distance traveled can be expressed as:
R = (r cos(β)) T
Substituting the expression for T we derived earlier:
R = (r cos(β)) * (2 r sin(β) / g)
Thus, the range becomes:
R = (2 r² sin(β) cos(β)) / g
Using Trigonometric Identities
We can simplify this further using the trigonometric identity:
sin(2β) = 2 sin(β) cos(β)
Therefore, the maximum range along the inclined plane can be expressed as:
R = (r² sin(2β)) / g
Final Thoughts
This formula gives us the maximum range of the bullet fired from the bottom of an inclined plane at an angle β with an initial velocity r. The key takeaway is that the range depends on both the initial velocity and the angle of inclination, showcasing the fascinating interplay between projectile motion and trigonometry.
