Projectile Motion Maximum Height And Initial Velocity Relationship

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When an object is thrown upwards with an initial velocity, its motion is governed by gravity. Understanding this relationship between initial velocity and maximum height is crucial in physics. This article delves into the physics behind projectile motion, specifically focusing on how doubling the initial velocity impacts the maximum height achieved by the object.

Key Concepts in Projectile Motion

To understand the problem, we need to grasp a few key concepts related to projectile motion:

  • Initial Velocity (u): This is the velocity with which the object is initially thrown upwards.
  • Final Velocity (v): At the maximum height, the object's velocity momentarily becomes zero before it starts falling back down. Therefore, the final velocity at the maximum height is 0.
  • Acceleration due to Gravity (g): This is the constant acceleration acting downwards on the object, approximately 9.8 m/s².
  • Maximum Height (h): This is the highest vertical distance the object reaches from its starting point.

Deriving the Formula for Maximum Height

We can use one of the equations of motion to relate these quantities. The relevant equation is:

v² = u² + 2as

Where:

  • v = final velocity
  • u = initial velocity
  • a = acceleration
  • s = displacement

In our case:

  • v = 0 (at maximum height)
  • a = -g (acceleration due to gravity, negative since it acts downwards)
  • s = h (maximum height)

Substituting these values into the equation, we get:

0 = u² - 2gh

Solving for h, we get the formula for maximum height:

h = u² / (2g)

Analyzing the Relationship Between Initial Velocity and Maximum Height

The formula h = u² / (2g) reveals a crucial relationship: the maximum height (h) is directly proportional to the square of the initial velocity (u²). This means if you double the initial velocity, the maximum height will increase by a factor of four. This square relationship is essential to understanding projectile motion problems. In essence, a small change in initial velocity can lead to a significant change in the maximum height reached.

Solving the Problem: Doubling the Initial Velocity

The original problem states that a body is thrown upwards with a velocity 'u' and reaches a maximum height 'h'. We are asked to find the new maximum height if the initial velocity is doubled (i.e., becomes 2u).

Using the formula h = u² / (2g), let's denote the initial maximum height as h₁ when the initial velocity is u:

h₁ = u² / (2g)

Now, let's calculate the new maximum height (h₂) when the initial velocity is doubled to 2u:

h₂ = (2u)² / (2g) h₂ = 4u² / (2g)

Comparing the New Height to the Original Height

To find the relationship between h₂ and h₁, we can divide the equation for h₂ by the equation for h₁:

h₂ / h₁ = (4u² / (2g)) / (u² / (2g)) h₂ / h₁ = 4

This result shows that h₂ = 4h₁. Therefore, if the initial velocity is doubled, the maximum height becomes four times the original height.

Why Does Doubling the Velocity Quadruple the Height?

This may seem counterintuitive at first. Why does doubling the velocity quadruple the height instead of just doubling it? The key lies in understanding the energy transformation involved in projectile motion. When an object is thrown upwards, its initial kinetic energy (energy of motion) is gradually converted into potential energy (energy of position due to height). The kinetic energy is proportional to the square of the velocity (KE = 1/2 * m * v²), and the potential energy at the maximum height is proportional to the height (PE = m * g * h). When you double the initial velocity, you quadruple the initial kinetic energy. This quadrupled energy is then converted into potential energy, resulting in a fourfold increase in the maximum height. Understanding the conservation of energy helps visualize why the relationship between initial velocity and maximum height is quadratic rather than linear.

Answering the Specific Question

The question asks: If its velocity of projection is doubled, the maximum height it reaches is:

Based on our calculations, the correct answer is:

  1. 4h

Common Mistakes and Misconceptions

A common mistake is assuming a linear relationship between initial velocity and maximum height. Many students might incorrectly think that doubling the velocity would simply double the height. It's crucial to remember the quadratic relationship derived from the equations of motion and the concept of energy conservation. Another misconception is neglecting the effect of gravity. Gravity is the constant force that decelerates the object as it moves upwards, ultimately bringing it to a momentary stop at its maximum height. Without considering gravity, the entire analysis of projectile motion would be flawed.

Extending the Concept: Applications and Further Exploration

Understanding the relationship between initial velocity and maximum height has numerous applications in real-world scenarios. For example, it's critical in sports like basketball, where players need to estimate the initial velocity and angle required to shoot the ball into the basket. It's also vital in engineering, such as designing projectiles or analyzing the trajectory of objects in various systems. Further exploration of projectile motion could involve considering air resistance, which adds complexity to the calculations but provides a more realistic model of motion in many situations. The study of projectile motion with air resistance often requires numerical methods and computer simulations.

Practice Problems

To solidify your understanding, try solving these practice problems:

  1. An object is thrown upwards with an initial velocity of 10 m/s and reaches a maximum height of 5 meters. What would be the maximum height if the initial velocity were 15 m/s?
  2. A ball is thrown upwards and reaches a maximum height of 20 meters. What was its initial velocity? (Assume g = 9.8 m/s²)
  3. Compare the maximum heights reached by two objects thrown upwards with initial velocities of 25 m/s and 30 m/s, respectively.

By working through these problems, you'll reinforce your grasp of the concepts and be better equipped to tackle similar challenges.

Conclusion: The Significance of the Quadratic Relationship

In conclusion, when a body is thrown upwards, doubling its initial velocity results in a fourfold increase in the maximum height it reaches. This quadratic relationship, h = u² / (2g), is a fundamental principle in physics, derived from the equations of motion and the concept of energy conservation. Understanding this relationship is essential for solving projectile motion problems and for analyzing real-world applications involving the motion of objects under the influence of gravity. Always remember to account for the crucial role of gravity and the energy transformations that occur throughout the projectile's trajectory. By mastering these concepts, you'll gain a deeper appreciation for the elegance and power of physics in describing the world around us.