If any ball projected from three different angles and cover same vertical velocity then their initial vertical velocity will be? Explain

When a ball is projected from different angles but covers the same vertical distance, the initial vertical velocity can be understood through the principles of projectile motion. Let's break this down step by step.

Understanding Vertical Velocity

Vertical velocity refers to the component of the ball's velocity that acts in the upward or downward direction. In projectile motion, this is influenced by the angle of projection and the initial speed of the ball.

Components of Velocity

When a ball is projected at an angle, its initial velocity can be divided into two components: vertical and horizontal. The vertical component can be calculated using the sine function:

• Vertical Velocity (Vy) = V * sin(θ)
• Horizontal Velocity (Vx) = V * cos(θ)

Here, V is the initial velocity, and θ is the angle of projection.

Same Vertical Distance, Different Angles

If three balls are projected from different angles but reach the same vertical height, it implies that their vertical components of velocity must be adjusted accordingly. The key point is that the maximum height achieved by each ball depends on its initial vertical velocity.

Calculating Initial Vertical Velocity

To achieve the same vertical height, the initial vertical velocities of the balls must be equal. This is because the maximum height (H) reached by a projectile can be calculated using the formula:

• H = (Vy²) / (2g)

Where g is the acceleration due to gravity (approximately 9.81 m/s²). Since the height is the same for all three balls, their vertical velocities must also be the same, regardless of the angle of projection.

Example Scenario

Imagine three balls projected at angles of 30°, 45°, and 60° with the same initial vertical velocity. If we denote this vertical velocity as Vy, we can express it as:

• For 30°: Vy = V * sin(30°) = V * 0.5
• For 45°: Vy = V * sin(45°) = V * √2/2
• For 60°: Vy = V * sin(60°) = V * √3/2

To ensure they all reach the same height, the initial vertical velocities must be adjusted based on the angle. However, if they are projected to reach the same height, we can conclude that the effective vertical component must be equal for all three angles.

Final Thoughts

In summary, when balls are projected from different angles but achieve the same vertical height, their initial vertical velocities must be equal. This equality allows them to reach the same maximum height despite the differences in their horizontal components and angles of projection. Understanding this relationship is crucial in the study of projectile motion and helps in predicting the behavior of objects in motion.