Calculate Trajectory of a Projectile

Projectile Motion Equations:

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

Initial Velocity (v₀):

m/s

Launch Angle (θ):

degrees

Time (t):

seconds

Gravity (g):

m/s²

Horizontal Position (x):

Vertical Position (y):

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1. What is Projectile Motion?

Projectile motion refers to the motion of an object thrown or projected into the air, subject only to acceleration due to gravity. The path followed by a projectile is called its trajectory, which is typically parabolic under ideal conditions.

2. How Does the Calculator Work?

The calculator uses the fundamental projectile motion equations:

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

Where:

\( x \) — Horizontal position (meters)
\( y \) — Vertical position (meters)
\( v_0 \) — Initial velocity (m/s)
\( \theta \) — Launch angle (degrees)
\( t \) — Time (seconds)
\( g \) — Acceleration due to gravity (9.81 m/s² on Earth)

Explanation: The horizontal motion is constant velocity (no acceleration), while vertical motion is affected by gravity.

3. Importance of Projectile Calculations

Details: Understanding projectile motion is essential in physics, engineering, ballistics, sports science, and many real-world applications like designing trajectories for rockets, artillery, or even sports equipment.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What assumptions does this calculator make?
A: It assumes no air resistance, flat ground, constant gravity, and that the projectile starts and lands at the same height.

Q2: How do I calculate maximum height?
A: Maximum height occurs when vertical velocity becomes zero. Use \( t = v_0 \sin(\theta)/g \) in the y equation.

Q3: How do I calculate range?
A: Range is the horizontal distance when y returns to zero. Total flight time is \( t = 2v_0 \sin(\theta)/g \).

Q4: What if my projectile lands at a different height?
A: The equations become more complex. You would need to solve the quadratic equation for when y equals the landing height.

Q5: Can I use this for objects with significant air resistance?
A: No, these equations don't account for air resistance. For objects like baseballs or bullets, more complex models are needed.