Find out time after which ball is moving perpendicular to it`s initial projection

To determine the time after which a ball is moving perpendicular to its initial projection, we need to analyze the motion of the ball in both horizontal and vertical directions. This involves understanding the principles of projectile motion, which can be broken down into two components: horizontal motion and vertical motion.

Understanding Projectile Motion

When a ball is projected at an angle, it follows a curved path known as a trajectory. The initial velocity can be split into two components:

• Horizontal Component (Vx): This remains constant throughout the motion since there is no horizontal acceleration (ignoring air resistance).
• Vertical Component (Vy): This changes due to the acceleration caused by gravity, which acts downwards at approximately 9.81 m/s².

Finding the Time for Perpendicular Motion

For the ball to be moving perpendicular to its initial projection, its velocity vector must be directed vertically. This means that the horizontal component of the velocity must be zero at that moment. However, since the horizontal velocity remains constant (Vx = V * cos(θ)), it will never actually become zero unless the ball hits the ground or is stopped by some other force. Instead, we can find the time when the vertical component of the velocity becomes equal to the horizontal component in magnitude but opposite in direction.

Mathematical Approach

Let’s denote:

• V: Initial velocity of the ball
• θ: Angle of projection
• g: Acceleration due to gravity (approximately 9.81 m/s²)

The horizontal and vertical components of the initial velocity can be expressed as:

• Vx = V * cos(θ)
• Vy = V * sin(θ) - g * t

To find the time when the ball's vertical velocity becomes equal in magnitude to the horizontal velocity, we set:

Vy = -Vx

Substituting the expressions for Vx and Vy, we get:

V * sin(θ) - g * t = -V * cos(θ)

Rearranging this equation gives:

g * t = V * sin(θ) + V * cos(θ)

Now, solving for time (t):

t = (V * (sin(θ) + cos(θ))) / g

Example Calculation

Let’s say a ball is projected with an initial velocity of 20 m/s at an angle of 45 degrees. We can calculate the time as follows:

• V = 20 m/s
• θ = 45 degrees (sin(45) = cos(45) = √2/2)
• g = 9.81 m/s²

Substituting these values into our time equation:

t = (20 * (√2/2 + √2/2)) / 9.81

t = (20 * √2) / 9.81

Calculating this gives approximately:

t ≈ 2.87 seconds

Final Thoughts

In this example, the ball will be moving perpendicular to its initial projection approximately 2.87 seconds after it was launched. This analysis highlights the importance of understanding both the horizontal and vertical components of motion in projectile dynamics. By breaking down the problem into manageable parts, we can effectively determine the time at which the ball's motion changes direction relative to its initial trajectory.