In physics, understanding the motion of projectiles is crucial for various applications, from sports to engineering. One common scenario involves determining the maximum height attained by a projectile launched at a specific angle with a given initial velocity. This article will walk you through the process using a scenario where a particle is projected at a 45-degree angle with a velocity of 30 m/s.

Understanding Projectile Motion

Projectile motion is a type of motion where an object is thrown/cannoned and moves under the influence of gravity. The motion of the projectile can be broken down into horizontal and vertical components. The key is to analyze these components separately to predict the motion accurately. In this discussion, we will focus on calculating the maximum height.

Formula and Calculation

The formula to determine the maximum height H of a projectile is given by:

Free Preparation: Open Google search for formula:

H u03C0u00b2 sin2θ / 2g

tH maximum height tu initial velocity 30 m/s tθ angle of projection 45 degrees tg acceleration due to gravity 9.8 m/s2

Step 1: Calculate sin2θ:

sin45° u221A2 / 2 u2248 0.707

Therefore, sin2 45° (0.707)2 u2248 0.5

Step 2: Substitute the values into the formula:

H (302 0.5) / (2 9.8)

H 900 0.5 / 19.6

H 450 / 19.6 u2248 22.95 meters

Thus, the maximum height attained by the particle is approximately 22.95 meters.

Solving a Similar Scenario

For further illustration, let's consider a scenario where a particle is projected at an angle of 60 degrees with an initial velocity of 50 m/s. The steps are similar:

tCalculate the vertical component of the initial velocity:

v0y v0 sinθ 50 sin60° 50 u221A3 / 2 u2248 43.30 m/s

Use the formula for maximum height:

H v0y2 / 2g

H (43.302) / (2 u22C5 9.81)

H 1875 / 19.62 u2248 95.66 meters

Therefore, the maximum height attained in this case is approximately 95.66 meters.

Conclusion

The process of finding the maximum height of a projectile involves breaking down the problem into manageable steps and using the appropriate formula. Whether it is a 45-degree projection with a velocity of 30 m/s or any other angle and velocity, the steps remain the same.

Understanding these principles not only helps in academic and competitive exams but also has real-world applications in fields such as sports, astronomy, and architecture.

References:

tPhysics Textbooks tOnline Physics Calculators tGoogle Search for Additional Formulas and Examples

Note: sin2θ and cos2θ are important trigonometric identities in physics.