In the figure shown, the two projectiles are fired simultaneously. What should be the initial speed of the left side projectile for the two projectiles to hit in mid-air?

To determine the initial speed of the left-side projectile so that both projectiles collide in mid-air, we need to analyze their motion using the principles of projectile motion. Let's break this down step by step.

Understanding Projectile Motion

Projectile motion can be described as the motion of an object that is launched into the air and is influenced only by the force of gravity (assuming air resistance is negligible). Each projectile follows a parabolic trajectory, and we can analyze their horizontal and vertical motions separately.

Key Variables

• Initial Velocity (u): The speed at which the projectile is launched.
• Angle of Projection (θ): The angle at which the projectile is fired.
• Time of Flight (t): The total time the projectile is in the air.
• Horizontal Distance (R): The range or distance traveled horizontally.
• Vertical Height (H): The maximum height reached by the projectile.

Equations of Motion

For any projectile, the horizontal and vertical motions can be described by the following equations:

• Horizontal motion: R = u * cos(θ) * t
• Vertical motion: H = u * sin(θ) * t - 0.5 * g * t²

Here, g is the acceleration due to gravity (approximately 9.81 m/s²).

Setting Up the Problem

Assuming both projectiles are fired at the same time and we know the angle of projection for both, we can set up the equations for their respective motions. Let’s denote:

• Projectile A (left side) with initial speed u_A and angle θ_A.
• Projectile B (right side) with initial speed u_B and angle θ_B.

Finding the Collision Point

For the projectiles to collide in mid-air, they must reach the same horizontal and vertical coordinates at the same time. This means:

• The horizontal distances covered by both projectiles must be equal at the time of collision.
• The vertical heights must also be equal at that same time.

Calculating the Initial Speed

Let’s assume that both projectiles are fired at angles that allow them to reach the same height at the same time. If we know the distance between the two launch points and the angles, we can derive the initial speed of the left-side projectile.

Using the horizontal motion equation, we can express the time of flight for both projectiles:

• For Projectile A: t = R_A / (u_A * cos(θ_A))
• For Projectile B: t = R_B / (u_B * cos(θ_B))

Setting these equal gives us a relationship between the initial speeds and the distances:

From this equation, we can solve for u_A if we know the values of u_B, R_A, R_B, θ_A, and θ_B.

Example Calculation

Suppose:

• u_B = 20 m/s
• R_A = 50 m
• R_B = 30 m

And both projectiles are fired at an angle of 45 degrees. Plugging these values into our derived equation will allow us to find the required initial speed for the left-side projectile.

Final Thoughts

By carefully analyzing the motion of both projectiles and applying the equations of projectile motion, you can determine the necessary initial speed for the left-side projectile to ensure they collide in mid-air. This approach not only helps in solving this specific problem but also enhances your understanding of the dynamics involved in projectile motion.