Projectile Motion Vertical Displacement Calculator

What is Projectile Motion Vertical Displacement?

Projectile motion vertical displacement is a fundamental concept in physics that describes the vertical distance traveled by an object launched into the air and moving along a parabolic path under the influence of gravity. It quantifies how far above or below its initial launch position the projectile has moved at any given moment during its flight.

Formula

The vertical displacement in projectile motion is given by the equation:

\[ y = v_0 \sin(\theta) t - \frac{1}{2}gt^2 \]

Where:

• \( y \) is the vertical displacement (in meters, m)
• \( v_0 \) is the initial velocity (in meters per second, m/s)
• \( \theta \) is the launch angle (in radians)
• \( t \) is the time elapsed since launch (in seconds, s)
• \( g \) is the acceleration due to gravity (approximately 9.8 m/s²)

Calculation Steps

Let's walk through a step-by-step calculation of vertical displacement:

1. Given:
   • Initial velocity (\( v_0 \)) = 20 m/s
   • Launch angle (\( \theta \)) = 45°
   • Time (\( t \)) = 2 s
   • Acceleration due to gravity (\( g \)) = 9.8 m/s²
2. Convert the angle to radians: \[ \theta = 45° \times \frac{\pi}{180°} = \frac{\pi}{4} \text{ radians} \]
3. Calculate \( \sin(\theta) \): \[ \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \approx 0.7071 \]
4. Apply the vertical displacement formula: \[ y = v_0 \sin(\theta) t - \frac{1}{2}gt^2 \] \[ y = 20 \times 0.7071 \times 2 - \frac{1}{2} \times 9.8 \times 2^2 \]
5. Perform the calculation: \[ y = 28.284 - 19.6 = 8.684 \text{ m} \]

Example and Visual Representation

Let's visualize the vertical displacement in projectile motion:

This diagram illustrates:

• The parabolic trajectory of the projectile (blue curve)
• The initial velocity vector (\( v_0 \)) at the launch point (red arrow)
• The vertical displacement (\( y \)) at t = 2s (yellow arrow)
• The launch angle (\( \theta \)) between the initial velocity and the horizontal
• The effect of gravity (\( g \)) pulling the projectile downward
• The position of the projectile at t = 2s (blue dot)