Understanding the Range of a Projected Body: A Detailed Analysis
Upon analyzing the question you posed regarding the motion of a projected body, it is essential to clarify the scenario and provide a comprehensive explanation. The problem enquires about the range of a body projected with a velocity of 20 m/s at an angle of 30 degrees with the horizontal. To address this, we will delve into the underlying physics, mathematical calculations, and the role of different parameters involved.
Background and Theory
In physics, the Trajectory of a projectile under the influence of gravity is a fascinating topic. The range of a projectile, defined as the horizontal distance it travels from the point of projection to the point of fall, is a critical parameter in understanding its motion. The range (R) can be calculated using the formula:
[ R frac{v^2 sin(2theta)}{g} ]
Where v is the initial velocity, theta is the angle of projection with the horizontal, and g is the acceleration due to gravity. Importantly, the term sin(2theta) reaches its maximum value when theta 45^circ. This is due to the trigonometric identity that the sine function reaches its peak value of 1 when its argument is 90 degrees, and thus sin(90^circ) 1.
Given Parameters and Calculations
In the specific scenario you provided, the velocity v 20 text{ m/s} and the angle of projection theta 30^circ. The acceleration due to gravity, g 9.81 text{ m/s}^2, is a standard physical constant. Applying these values to the range formula, we get:
[ R frac{(20)^2 sin(2 times 30^circ)}{9.81} ]
[ sin(60^circ) frac{sqrt{3}}{2} approx 0.866 ]
[ R frac{400 times 0.866}{9.81} approx 35.00 text{ meters} ]
Factors Affecting the Range
Several factors can influence the range of a projected body, including the initial velocity, angle of projection, and the gravitational acceleration:
Initial Velocity: Higher initial velocity results in a longer range. In our example, if the initial velocity were increased to 40 text{ m/s}, the range would be doubled. Angle of Projection: The optimal angle to achieve maximum range, as mentioned earlier, is 45 degrees. However, for angles other than 45 degrees, the range is determined by the sine of twice the angle. Therefore, for the given angle of 30 degrees, the range calculated is correct and not a "maximum range" but rather a specific range for those parameters. Gravitational Acceleration: In different locations on Earth, the gravitational acceleration varies slightly, which can affect the range calculation. On the surface, it is generally assumed to be 9.81 m/s2.Conclusion
In conclusion, the range of a projected body is not a "maximum range" unless the angle of projection is 45 degrees. In the given scenario with a velocity of 20 m/s at 30 degrees, the range is approximately 35 meters. Understanding these principles is essential for a wide range of applications, from sports and engineering to space exploration.
To further explore this topic, you may consider experimenting with different angles and velocities, or investigating the effects of air resistance and other factors that can influence projectile motion. The knowledge of projectile motion is foundational in many branches of physics and has practical applications in everyday life.