Find The Height Of The Tower And The Initial Velocity With Which The Body Was Thrown Horizontally From The Top Of The Tower, Given That It Strikes The Ground After 3 Seconds At An Angle Of 45 Degrees With The Horizontal.

by ADMIN 221 views

In the captivating realm of physics, projectile motion stands as a cornerstone concept, illustrating the graceful dance of objects influenced solely by gravity after their initial impetus. This article delves into a quintessential projectile motion problem, dissecting the trajectory of a body hurled horizontally from a tower's summit. By meticulously analyzing the given parameters – the time of flight and the impact angle – we shall unravel the tower's height and the body's initial projection speed. This exploration will not only solidify your understanding of projectile motion principles but also equip you with the problem-solving prowess to tackle similar scenarios.

Problem Statement: A Horizontal Launch

Imagine a scenario where an object is launched horizontally from the apex of a tower. This projectile embarks on its journey, gracefully arcing through the air under the constant tug of gravity. After precisely 3 seconds, it culminates its flight by striking the ground. Intriguingly, the impact occurs at an angle of 45 degrees with respect to the horizontal plane. Our mission is twofold: to determine the height of the tower from which the projectile was launched and to ascertain the initial speed with which the object was projected into its trajectory. This problem encapsulates the essence of projectile motion, inviting us to dissect the interplay between horizontal and vertical motion components.

Decoding the Physics: Projectile Motion Principles

Before we embark on the calculations, let's reinforce the fundamental principles governing projectile motion. The trajectory of a projectile, in the absence of air resistance, is a graceful parabola. This curved path arises from the projectile's constant horizontal velocity, coupled with its uniformly accelerated vertical motion due to gravity. The key to unraveling projectile motion problems lies in treating the horizontal and vertical components of motion independently. This separation allows us to apply the appropriate kinematic equations to each direction, paving the way for a clear and methodical solution.

  • Horizontal Motion: The horizontal velocity of the projectile remains constant throughout its flight, assuming negligible air resistance. This is because there is no horizontal force acting upon it. The horizontal distance traveled is simply the product of the initial horizontal velocity and the time of flight.
  • Vertical Motion: The vertical motion is governed by gravity, resulting in a constant downward acceleration (approximately 9.8 m/s² on Earth). The vertical velocity changes linearly with time, and the vertical displacement is determined by the kinematic equations of motion under constant acceleration.

Part (i): Unveiling the Tower's Height

Our first objective is to determine the height of the tower. Since the projectile is launched horizontally, its initial vertical velocity is zero. The vertical motion is solely influenced by gravity. We can employ the following kinematic equation to relate the vertical displacement (tower height, h), initial vertical velocity (v₀y), acceleration due to gravity (g), and time (t):

h = v₀y * t + (1/2) * g * t²

In this scenario, v₀y = 0 m/s (initial vertical velocity), g = 9.8 m/s² (acceleration due to gravity), and t = 3 s (time of flight). Substituting these values into the equation, we get:

h = 0 * 3 + (1/2) * 9.8 * (3)²

h = 0 + (1/2) * 9.8 * 9

h = 44.1 meters

Therefore, the height of the tower is 44.1 meters. This calculation showcases the direct application of kinematic equations to determine the vertical displacement under constant acceleration due to gravity.

Part (ii): Deciphering the Initial Projection Speed

Now, let's turn our attention to the second part of the problem: determining the initial speed with which the body was projected. We are given that the projectile strikes the ground at an angle of 45 degrees with the horizontal. This crucial piece of information allows us to relate the horizontal and vertical components of the final velocity.

At the point of impact, the projectile possesses both horizontal (vx) and vertical (vy) velocity components. The angle of 45 degrees implies that the magnitudes of these components are equal:

|vx| = |vy|

This equality stems from the fact that the tangent of 45 degrees is 1, and the tangent of the angle of impact is the ratio of the vertical velocity component to the horizontal velocity component.

To find the vertical velocity component (vy) at impact, we can use another kinematic equation:

vy = v₀y + g * t

Again, v₀y = 0 m/s, g = 9.8 m/s², and t = 3 s. Plugging in the values, we obtain:

vy = 0 + 9.8 * 3

vy = 29.4 m/s

Since |vx| = |vy|, the magnitude of the horizontal velocity component is also 29.4 m/s. Recall that the horizontal velocity remains constant throughout the projectile's flight. Therefore, the initial horizontal velocity (v₀x) is equal to the horizontal velocity at impact:

v₀x = vx = 29.4 m/s

The initial projection speed is simply the magnitude of the initial horizontal velocity, as the projectile was launched horizontally:

Initial Speed = v₀x = 29.4 m/s

Thus, the body was projected with an initial speed of 29.4 meters per second. This part of the solution highlights the importance of understanding the relationship between velocity components and the angle of impact in projectile motion problems.

Synthesis: Key Takeaways and Insights

In this exploration of projectile motion, we successfully determined both the height of the tower (44.1 meters) and the initial speed of the projectile (29.4 m/s). The solution hinged on the fundamental principles of projectile motion: the independence of horizontal and vertical motion, the constant horizontal velocity, and the uniformly accelerated vertical motion due to gravity. By applying the appropriate kinematic equations and leveraging the information provided about the impact angle, we were able to systematically unravel the problem.

This example underscores the power of analytical thinking in physics. By breaking down a complex problem into simpler components and applying well-established principles, we can arrive at elegant and accurate solutions. Projectile motion problems, like this one, serve as valuable exercises in developing problem-solving skills and deepening our understanding of the physical world.

Further Exploration: Extending the Concepts

To further solidify your grasp of projectile motion, consider exploring variations of this problem. For instance, you could investigate scenarios where the projectile is launched at an angle to the horizontal, introducing an initial vertical velocity component. You could also explore the effects of air resistance on the projectile's trajectory, adding another layer of complexity to the analysis. These extensions will challenge your understanding and broaden your perspective on the fascinating world of projectile motion.

By mastering the concepts presented in this article and engaging in further exploration, you will be well-equipped to tackle a wide range of projectile motion problems and appreciate the beauty and elegance of physics in action. Remember, the key to success lies in a clear understanding of the fundamental principles and a methodical approach to problem-solving.