At what angle to the horizontal must the nozzle of a machine gun be kept when firing to obtain a maximum horizontal range for the bullet(show your working)
the maximum angle is 45 degrees
To determine the angle at which the nozzle of a machine gun must be kept to obtain a maximum horizontal range for the bullet, we can use principles of projectile motion. Let's assume there is no air resistance.
The maximum horizontal range is achieved when the projectile covers an equal distance in the upward and downward directions. This occurs when the projectile is launched at an angle of 45 degrees to the horizontal. To understand why, let's break down the projectile's motion:
1. The initial velocity (V₀) of the projectile can be resolved into horizontal (V₀x) and vertical (V₀y) components.
2. The horizontal component (V₀x) remains constant throughout the projectile's flight since there are no forces acting on it horizontally. Thus, the time of flight is the same for both the upward and downward trajectories.
3. The vertical component (V₀y) follows a parabolic path due to the force of gravity, reaching its highest point halfway through the time of flight. At this highest point, the vertical component of velocity becomes zero.
4. Assuming symmetrical trajectories, the time taken to reach the maximum height will be half of the total time of flight.
5. As the projectile comes down from its maximum height, the vertical component (V₀y) accelerates downward due to gravity.
6. When the projectile reaches the same height from where it was launched, it will have the same magnitude but opposite direction for its vertical component. At this point, the vertical component of velocity will be equal to the initial velocity (V₀y) but in the opposite direction.
7. Finally, the projectile hits the ground at a distance equal to the range of the maximum height achieved.
Since the horizontal component of velocity remains constant during the entire flight, the time taken to reach the maximum height will be half of the total time of flight. Therefore, the angle required for maximum range occurs when the time taken for the upward trajectory equals the time taken for the downward trajectory, which is achieved at an angle of 45 degrees.
In summary, the nozzle of the machine gun should be kept at an angle of 45 degrees to the horizontal to achieve the maximum horizontal range for the bullet.