Projectile motion horizontal displacement refers to the distance a projectile travels horizontally from its starting point to its landing point. It's a key concept in physics, particularly in the study of motion and trajectories. This displacement is influenced by the initial velocity, launch angle, and time of flight, assuming no air resistance.

The formula for calculating horizontal displacement in projectile motion is:

\[ d = v_0 \cos(\theta) t \]

Where:

- \( d \) is the horizontal displacement (in meters)
- \( v_0 \) is the initial velocity (in meters per second)
- \( \theta \) is the launch angle (in radians)
- \( t \) is the time of flight (in seconds)

Let's calculate the horizontal displacement for a typical projectile motion scenario:

- Given:
- Initial velocity (\( v_0 \)) = 20 m/s
- Launch angle (\( \theta \)) = 30°
- Time of flight (\( t \)) = 2 s

- Convert the angle to radians: \[ 30° \times \frac{\pi}{180°} = 0.5236 \text{ radians} \]
- Calculate the horizontal displacement: \[ d = 20 \times \cos(0.5236) \times 2 \] \[ d = 20 \times 0.8660 \times 2 \] \[ d = 34.64 \text{ meters} \]

Let's visualize the projectile motion and horizontal displacement:

This diagram illustrates:

- The parabolic path of the projectile (blue curve)
- The launch point (green dot) and landing point (red dot)
- The launch angle θ (yellow arc)
- The initial velocity vector v₀ (teal arrow)
- The horizontal displacement, which is the distance between the launch and landing points along the ground

Understanding horizontal displacement in projectile motion is crucial for various applications, including sports (e.g., long jump, javelin throw), ballistics, and engineering design of projectile-based systems.

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