Critical Thinking – Creative Exercise
Besieging a City
You are the head engineer of a medieval army, currently besieging a walled city (see figure). Orders have come from the commander directing you to construct an instrument of war, designed to attack the enemy from outside of the walls. Catapults and trebuchets have not been invented yet, so you decide to build a giant slide with a ramp that will launch a projectile with sufficient fury to clear the wall and land among the enemy king’s tents, instantly demoralizing his army and ensuring a swift victory. To keep your men and the slide safe, it must remain a distance 𝑑=100 m away from the wall, out of the range of enemy archers. The king has set his tents 𝑅=30 m beyond the wall, which is 𝐻=30 m high.
Your slide has two parameters that must be determined, the height (h) and the launch angle (θ). They must be chosen so that the projectile clears the wall and lands on the king. Assume the projectile leaves from ground level. Ignore friction and air resistance.
Choose the values h and θ such that your slide wins the war.
Before you invest resources constructing the slide you need to analyze the situation in order to be sure it will meet its requirements. What are the criteria for your weapon to be successful (there are two)? Does it meet them?
Preliminary considerations (do/answer them):
1) This is a complex problem. Often, it is helpful to break complex problems into pieces and then consider the pieces individually. Once you understand the pieces, you can try to connect them together. We can do such things to this problem. What are the pieces? What concepts are relevant in each piece?
2) Sometimes, it’s also helpful to first consider an easier problem. Here’s one for the current problem (work on it first): An object is launched from the ground at an angle θ. Find the speed necessary for it to land an arbitrary distance L away (ignore the wall).
3) Try to proceed systematically. What is the next complication to add to the problem? {Just consider this question. No need to include a response in your answer.}
To answer:
1) What are h and θ for your slide?
2) What is the speed, vo, with which your projectile enters the air?
3) What are the criteria for success? Show that your slide meets them.

To tackle the challenge of designing a slide that can launch a projectile over a wall and hit the enemy king's tents, we need to break down the problem into manageable parts. This involves understanding the physics of projectile motion, determining the necessary parameters for the slide, and ensuring that our design meets the criteria for success. Let's dive into this step by step.

Defining Success Criteria

Before we can determine the height (h) and launch angle (θ) for our slide, we need to establish what makes our weapon successful. There are two primary criteria:

• Clear the Wall: The projectile must have enough vertical height to surpass the wall, which is 30 meters high.
• Hit the Target: The projectile must land at least 30 meters beyond the wall, reaching a total horizontal distance of 130 meters from the launch point.

Breaking Down the Problem

To find suitable values for h and θ, we can break the problem into smaller components:

• Projectile Motion Basics: We need to understand the equations governing projectile motion, specifically how height and distance relate to launch angle and initial speed.
• Vertical Motion: We will analyze how high the projectile must go to clear the wall.
• Horizontal Motion: We will determine how far the projectile travels horizontally based on the launch angle and initial speed.

Starting with a Simplified Problem

Let’s first consider a simpler scenario: launching a projectile from ground level to land at a distance L without any obstacles. The horizontal distance (L) can be expressed using the formula:

L = (v₀² * sin(2θ)) / g

Where:

• v₀: Initial speed of the projectile
• g: Acceleration due to gravity (approximately 9.81 m/s²)

Finding the Required Speed and Angles

Now, let’s apply this to our specific case. We want the projectile to land 130 meters away, so we set L = 130 m. Rearranging the formula gives us:

v₀² = (L * g) / sin(2θ)

Calculating the Launch Angle

To ensure the projectile clears the wall, we need to find the height it reaches at the horizontal distance of 100 m (the distance from the slide to the wall). The height (y) at any distance (x) can be described by:

y = x * tan(θ) - (g * x²) / (2 * v₀² * cos²(θ))

Setting y ≥ 30 m when x = 100 m gives us a condition to satisfy. We can use trial and error or numerical methods to find suitable values for θ and v₀ that meet both criteria.

Example Calculation

Let’s assume an angle θ of 45 degrees, which is often optimal for maximizing range. Plugging this into our equations:

• sin(90°) = 1
• cos(45°) = √2/2

Using the range equation:

v₀² = (130 m * 9.81 m/s²) / 1 = 1275.3 m²/s²

Thus, v₀ ≈ 35.7 m/s.

Verifying the Height Condition

Now we check if this speed and angle allow the projectile to clear the wall:

y = 100 * tan(45°) - (9.81 * 100²) / (2 * (35.7)² * (0.707)²)

Calculating this gives us:

y = 100 - (98100 / 2 * 1275.3) ≈ 100 - 38.5 ≈ 61.5 m

This height is sufficient to clear the 30 m wall.

Final Parameters

In summary, the parameters for our slide are:

• Height (h): The slide can be designed to be at least 30 m high to ensure the projectile clears the wall.
• Launch Angle (θ): An angle of 45 degrees is optimal for maximizing range.
• Initial Speed (v₀): Approximately 35.7 m/s is required to ensure the projectile lands 130 m away.

By carefully analyzing the physics of projectile motion and ensuring our design meets the criteria for success, we can confidently proceed with constructing the slide to secure victory in the siege.