Projectile motion vertical velocity is the component of velocity in the vertical direction for an object launched into the air and moving under the influence of gravity. It changes continuously during the projectile's flight due to the constant acceleration of gravity. Understanding vertical velocity is crucial for analyzing the motion of projectiles, such as balls, rockets, or any object thrown or launched into the air.

The formula for vertical velocity in projectile motion is:

\[ v_y = v_0 \sin(\theta) - gt \]

Where:

- \( v_y \) is the vertical velocity at time \( t \) (in m/s)
- \( v_0 \) is the initial velocity (in m/s)
- \( \theta \) is the launch angle (in radians)
- \( g \) is the acceleration due to gravity (approximately 9.81 m/s²)
- \( t \) is the time elapsed since launch (in seconds)

Let's calculate the vertical velocity for a projectile:

- Given:
- Initial velocity (\( v_0 \)) = 50 m/s
- Launch angle (\( \theta \)) = 60°
- Time (\( t \)) = 2 s
- Acceleration due to gravity (\( g \)) = 9.81 m/s²

- Convert the launch angle to radians: \[ \theta = 60° \times \frac{\pi}{180°} \approx 1.0472 \text{ radians} \]
- Calculate \( v_0 \sin(\theta) \): \[ v_0 \sin(\theta) = 50 \text{ m/s} \times \sin(1.0472) \approx 43.30 \text{ m/s} \]
- Calculate \( gt \): \[ gt = 9.81 \text{ m/s}^2 \times 2 \text{ s} = 19.62 \text{ m/s} \]
- Apply the vertical velocity formula: \[ v_y = 43.30 \text{ m/s} - 19.62 \text{ m/s} = 23.68 \text{ m/s} \]

Let's visualize the vertical velocity in projectile motion:

This diagram illustrates:

- The parabolic trajectory of the projectile (blue curve)
- The projectile at a specific point in its flight (yellow circle)
- The total velocity vector (red arrow)
- The vertical velocity component \( v_y \) (teal dashed arrow)
- The ground level (green line)

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