Projectile motion problem. The maximum range for a projectile is achieved when the projectile is fired at 45 degrees. This is true if the launch starts and ends at the same altitude.
What about if you fire at a target at a lower elevation? Is the optimal angle still 45 degrees? Is it more? Less?
Please help me explain in words not equations.
optimal angle same.
why> range depends on time in air, and horizontal velocity. If you solve for max time in air as a function of launch angle, and then horizontal velocity as a function of angle, and multiply them, you find that range as a function of angle is 2*cosAngle*sinAngle. Trig, indicates that is equivalent to sin^2(2*Angle )
Well sin of any angle is max when the angleis 90 deg, so for sin^2(2*launchA) to be max, launch angle is 45deg
When the projectile is fired at a target at a lower elevation, the optimal angle for achieving the maximum range is no longer 45 degrees. In fact, the optimal angle will be less than 45 degrees.
To understand why, let's consider the factors that affect the range of a projectile. The range is determined by the initial velocity of the projectile and the angle at which it is launched. When the launch and landing altitudes are the same, the angle that maximizes the range is 45 degrees because it allows for the greatest horizontal displacement.
However, when the target is at a lower elevation, the downward pull of gravity will have a greater effect on the projectile's trajectory. This means that the projectile will spend more time in the air, and therefore, it will have a longer hang time. As a result, the best angle to achieve the maximum range is less than 45 degrees.
At lower launch angles, the vertical component of the initial velocity becomes more significant, allowing the projectile to spend more time in the air and cover a greater horizontal distance. Therefore, the optimal angle for maximum range decreases when the target is at a lower elevation.
In conclusion, when firing at a target at a lower elevation, the optimal angle for maximum range is less than 45 degrees. The exact angle will depend on factors such as the target distance, projectile velocity, and the difference in elevation between the launch and target points.
When considering projectile motion, the maximum range is achieved when the launch angle is 45 degrees and the starting and ending points are at the same altitude. This means that the projectile is fired in such a way that it travels the farthest horizontally.
However, if you are firing at a target at a lower elevation, the optimal launch angle for maximum range would be less than 45 degrees. This is because when you aim at a lower target, a higher launch angle would cause the projectile to travel higher in the air but also come down steeper and reach the target sooner, resulting in a shorter horizontal distance.
To explain it in simple terms, imagine throwing a ball towards a lower target. If you throw it at a very high angle, it will go up high in the air and come down at a steep trajectory, hitting the target at a short horizontal distance. On the other hand, if you throw it at a lower angle, it will have a more shallow trajectory, reaching a higher horizontal distance before hitting the target.
Therefore, when aiming at a target at a lower elevation, the optimal launch angle for maximum range would be less than 45 degrees. The specific optimal angle would depend on factors such as the vertical distance between the starting and ending points, the initial velocity of the projectile, and the gravitational force acting on it.