Vertical Projectile Motion Calculator

Vertical Displacement Equation:

\[ D_y = V \sin(\theta) t - \frac{1}{2} g t^2 \]

Initial Velocity (V):

m/s

Launch Angle (θ):

degrees

Time (t):

seconds

Gravity (g):

m/s²

Vertical Displacement (D_y):

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1. What is Vertical Projectile Motion?

Vertical projectile motion describes the movement of an object launched into the air and moving under the influence of gravity. The vertical displacement depends on initial velocity, launch angle, time, and gravitational acceleration.

2. How Does the Calculator Work?

The calculator uses the vertical displacement equation:

\[ D_y = V \sin(\theta) t - \frac{1}{2} g t^2 \]

Where:

\( D_y \) — Vertical displacement (m)
\( V \) — Initial velocity (m/s)
\( \theta \) — Launch angle (degrees)
\( t \) — Time (seconds)
\( g \) — Gravitational acceleration (m/s²)

Explanation: The equation calculates the vertical position of a projectile at any given time, accounting for both the initial upward velocity component and the downward acceleration due to gravity.

3. Applications of Projectile Motion

Details: Projectile motion calculations are essential in physics, engineering, sports science, ballistics, and many other fields where objects are launched or thrown.

4. Using the Calculator

Tips: Enter initial velocity in m/s, launch angle in degrees (0-90), time in seconds, and gravitational acceleration (default is Earth's gravity 9.81 m/s²). All values must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What happens if the result is negative?
A: A negative result indicates the projectile is below the launch point, which typically occurs after reaching maximum height and descending.

Q2: What is the maximum height of a projectile?
A: Maximum height occurs when vertical velocity becomes zero and can be calculated using \( h_{max} = \frac{(V \sin(\theta))^2}{2g} \).

Q3: How does launch angle affect the trajectory?
A: A 45° angle typically gives maximum range, while a 90° angle gives maximum height but no horizontal displacement.

Q4: Does air resistance affect these calculations?
A: Yes, this equation assumes no air resistance. In reality, air resistance reduces both range and height of projectiles.

Q5: Can I use this for objects launched from a height?
A: This equation calculates displacement from the launch point. For objects launched from height, you would need to add the initial height to the result.